Futures margin modeling system

ABSTRACT

A clearinghouse computing device may be configured to generate a margin requirement for a portfolio of financial products and may include a processor to process instructions that cause the clearinghouse computing device to retrieve a plurality of pricing records from a historical pricing database, process the plurality of pricing records to generate rolling time series pricing records for at least one financial product having a plurality of dimensions, reduce the number of dimensions from a starting dimension to a reduced dimension, perform variance scaling and correlation scaling on the reduced dimension rolling time series pricing records, and generate a margin requirement based on a value-at-risk calculation.

REFERENCE TO RELATED APPLICATIONS

This application is a continuation under 37 C.F.R. § 1.53(b) of U.S.patent application Ser. No. 15/001,997 filed Jan. 20, 2016 (AttorneyDocket No. 004672-15515Z-US)) now U.S. Pat. No. ______, the entiredisclosure of which is hereby incorporated by reference and relied upon.

FIELD OF THE INVENTION

Embodiments of the present disclosure relate to systems and methods forprocessing data. More particularly, the invention provides computingsystems for processing data for modeling and determining marginrequirements.

BACKGROUND

Futures contracts and options on futures contracts can be distributed orexecuted through a variety of means. Historically, futures were largelyexecuted or transacted on the floor of an exchange in so-called “tradingpits” that facilitated physical interaction between floor brokers andfloor traders. This method is commonly known as “open outcry.” Althoughsome open outcry trading still occurs, most futures contracts andoptions are now transacted through electronic trading systems. Theseelectronic trading systems allow customers (e.g., parties wishing totransact in futures contracts and/or options) to establish an electroniclink to an electronic matching engine of a futures exchange. Thatengine, which may be implemented as a specially programmed computersystem or as part of a larger specially programmed electronic tradingcomputer system, may identify orders that match in terms of commodity,quantity and price.

Clearinghouses and other entities that clear trades require traders,such as traders of futures contracts, to maintain performance bonds inmargin accounts to cover risks associated with the portfolios. Theclearinghouse (e.g., central counterparty to financial products) may usethe performance bond to counter margin risk associated with theportfolio. Risks may utilize complex algorithms to be analyzed todetermine required initial margin amounts and maintenance marginamounts. A risk calculation module (or risk processor) may assist in thecalculation. In some examples, values (e.g., swap DV01s, volatilityvalues, etc.) and adjustments/factors (e.g., calendar chargeadjustments, liquidity charge minimums, etc.) may be used to enhance themargin calculation.

Clearinghouses are structured to provide exchanges and other tradingentities with solid financial footing. Maintaining proper margin amountsis an important part of the maintaining solid financial footing. Therequired margin amount generally varies according to the volatility of afinancial instrument; the more volatility, the larger the requiredmargin amount. This is to ensure that the bond will sufficiently coverthe cumulated losses that a contract would likely incur over a giventime period, such as a single day. Required margin amounts may bereduced where traders hold opposite positions in closely correlatedmarkets or spread trades.

Calculating margin amounts can be a challenge, even when computerdevices are utilized. In the trading environment the speed with whichinformation can be determined and distributed to market participants canbe critical. For example, regulations set time limits for clearingentities to provide margin requirements to market participants after theend of a trading day. Some market participants also expect clearingentities to quickly determine how a potential transaction will impacttheir margin requirements.

As the numbers of accounts and transactions increase over a larger fieldof trading products, it becomes difficult for existing computer systemsand processes to determine and communicate pricing, volatility andmargin requirements to market participants in the time frames requiredby regulations or expected by the market participants. Therefore thereis a need in the art for more efficient computer systems andcomputer-implemented methods for processing data to model and determinemargin requirements.

SUMMARY

This Summary is provided to introduce a selection of concepts in asimplified form that are further described below in the DetailedDescription. This Summary is not intended to identify key or essentialfeatures of the invention.

In at least some embodiments, a clearinghouse computing device may beconfigured to generate a margin requirement for a portfolio of financialproducts and may include a processor to process instructions that causethe clearinghouse computing device to retrieve a plurality of pricingrecords from a historical pricing database, process the plurality ofpricing records to generate rolling time series price daily log changesfor at least one financial product having a plurality of dimensions,reduce the number of dimensions from the initial dimension to a reduceddimension by mapping the rolling time series price daily log changes toa set of new risk factors, perform variance scaling and covariancescaling on the reduced dimension risk factor time series to generatescenarios, map the reduced dimension risk factor scenarios back to theinitial dimension price daily log change scenarios, and generate amargin requirement based on a value-at-risk calculation.

Embodiments include, without limitation, methods for spot trading,methods for support of spot trading by an exchange trading in futurescontracts and/or futures contract options, computer systems configuredto perform such methods, and computer-readable media storinginstructions that, when executed, cause a computer system to performsuch methods.

BRIEF DESCRIPTION OF THE DRAWINGS

Some embodiments are illustrated by way of example, and not by way oflimitation, in the figures of the accompanying drawings and in whichlike reference numerals refer to similar elements:

FIG. 1 shows an illustrative trading network environment forimplementing trading systems and methods according to aspects of thedisclosure;

FIG. 2 shows a block diagram representation of a clearinghouse computingsystem for calculating a margin requirement according to aspects of thedisclosure;

FIG. 3 shows an illustrative method performed using computer-executableinstructions when processed by a clearinghouse computing systemaccording to aspects of this disclosure;

FIG. 4 shows an illustrative listing of futures products correspondingto a corresponding commodity according to aspects of the disclosure;

FIG. 5 shows an illustrative chart depicting illustrative time series ofsettlement pricing information of a futures product having differentmaturity dates according to aspects of the disclosure;

FIG. 6 shows an illustrative two-sided KS test to analyzetime-to-maturity effects on a price daily log return distributionaccording to aspects of the disclosure;

FIGS. 7 and 8 each show an illustrative charts showing a distribution ofsamples used in the two-sided KS test for different particularcommodities according to aspects of the disclosure;

FIGS. 9-11 show illustrative log-return correlations of the front monthcontract to the second month contract using an exponential decay modelfor different financial products according to aspects of thisdisclosure;

FIGS. 12 shows illustrative charts showing traded volume curves for 6consecutive contracts associated with a particular commodity and adistribution of crossing days for the front and second months contractsaccording to aspects of the disclosure;

FIG. 13 shows illustrative charts showing open interest curves for 6consecutive contracts associated with the particular commodity and adistribution of crossing days for the front and second months contractsaccording to aspects of the disclosure;

FIG. 14 shows an illustrative table showing traded volume analysisassociated with a plurality of financial products, according to aspectsof the disclosure;

FIG. 15 shows an illustrative table showing open interest statisticsassociated with a plurality of financial products, according to aspectsof the disclosure;

FIG. 16 shows an illustrative chart showing results of aKolmogorov-Smirnov test for each of a plurality of interpolation methodsaccording to aspects of this disclosure;

FIGS. 17-19 show illustrative histograms for the return distributions aplurality of interpolation methods according to aspects of thisdisclosure;

FIG. 20 shows an illustrative result table for various commoditiesaccording to aspects of the disclosure;

FIG. 21 shows illustrative chart showing an example for the performanceof the spline interpolation and PCHIP for coal price data according toaspects of the disclosure;

FIG. 22 illustrates a schematic explanation of the data input into thetime series extension according to aspects of the disclosure;

FIGS. 23-26 show illustrative charts showing plots of real versusextended volatility scaled price daily log returns of futures productsaccording to aspects of the disclosure;

FIG. 27 shows an illustrative chart showing explained variance perprincipal component (PC) with Principal Component Analysis (PCA), adimension reduction model using orthogonal components, for crude oilaccording aspects of the disclosure;

FIG. 28 shows an illustrative chart showing explained variance for athreshold selection criteria for a financial product according toaspects of the disclosure;

FIG. 29 shows an illustrative chart showing a number of selected PCsneeded to reach the variance explained threshold for a financial productaccording to aspects of the disclosure;

FIG. 30 shows an illustrative chart showing explained variance usingconstantly three factors for a financial product according to aspects ofthe disclosure;

FIG. 31 shows an illustrative chart that represents the time-series ofthe leading three PC scores of a financial product according to aspectsof the disclosure;

FIG. 32 shows an illustrative table corresponding to visual inspectionof PC surface stability for the different curves according to aspects ofthe invention

FIGS. 33 and 34 show illustrative charts showing the insignificance ofthe autocorrelation and the significance of volatility clustering of theselected PC scores for a financial product according to aspects of thedisclosure;

FIGS. 35A-35C show a visual representation of an inspection of PCsurface stability for a plurality of curves according to aspects of thedisclosure.

FIG. 36 shows an illustrative chart showing dynamic correlation betweenPC scores using updated PC associated with a financial product accordingto aspects of the disclosure;

FIG. 37 shows an illustrative chart showing dynamic correlation betweenPC scores using non-updated PC corresponding to a financial productaccording to aspects of the disclosure;

FIG. 38 illustrates and quantities the volatility clustering of theresiduals, i.e. the non-selected PC scores according to aspects of thedisclosure;

FIG. 39 shows a graphical representation of the residuals (e.g., thenon-selected PC score) time series for an orthogonal model for aparticular underlying commodity of the financial product according toaspects of the disclosure;

FIGS. 40 and 41 show illustrative representations of the residualsurface with the 2-Factor model, an alternative dimension reductionmethod for PCA, for crude oil with and without regularization accordingto aspects of the disclosure;

FIG. 42 shows an illustrative chart showing a time evolution of thesecond parameter of the 2-Factor model without regularizations accordingto aspects of the disclosure;

FIG. 43 shows an illustrative chart showing a time evolution of thesecond parameter of the 2-Factor model using the regularizationaccording to aspects of the disclosure;

FIG. 44 shows an illustrative chart showing autocorrelation for thefirst parameter changes of the 2-Factor model system according toaspects of the disclosure;

FIG. 45 shows an illustrative chart showing autocorrelation for thefirst innovation, which is obtained by filtering the first parameterchanges by a dynamic autoregressive and volatility model, of the2-Factor model using regularization, according to aspects of thedisclosure;

FIG. 46 shows an illustrative chart showing results of a violationcoverage test for the 2-Factor model according to aspects of thedisclosure;

FIG. 47 shows an illustrative residual surface for a financial productwith the polynomial model, another alternative dimension reductionmethod for PCA, and without regulation according to aspects of thedisclosure;

FIG. 48 shows an illustrative chart of a time series of the firstparameter for the polynomial model without regularization according toaspects of the disclosure;

FIG. 49 shows an illustrative chart plotting the volatility of the firsttwo PC scores with EWMA method, a dynamic volatility model, for afinancial product according to aspects of this disclosure.

FIG. 49A shows an illustrative plot of the squared innovations derivedfrom the first principle component (PC);

FIG. 50 shows an illustrative chart showing a comparison between EWMAmodel and one of its alternative methods on the first PC score for afinancial product according to aspects of the disclosure;

FIG. 50A shows an illustrative chart plotting illustrative parametersfor a fast and slow dynamic model associated with a dual-lambda approachfor the first score of crude oil;

FIG. 51 shows illustrative charts showing the historical value of thefloor through time for a financial product, according to aspects of thisdisclosure;

FIG. 52 shows an illustrative chart showing historical dynamic floor fordifferent quantile for a financial product, when a standard EWMA isapplied as volatility model according to aspects of this disclosure;

FIGS. 53A and 53B show an illustrative comparison of the resultingcorrelation using EWMC model against DCC model of the first generics oftwo commodities;

FIG. 54 shows test success criteria for some statistical tests onbacktesting results according to aspects of the disclosure; and

FIG. 55 shows a number of margin violations in a one year moving windowfor a financial product according to aspects of the disclosure.

DETAILED DESCRIPTION

In the following description of various embodiments, reference is madeto the accompanying drawings, which form a part hereof, and in whichvarious embodiments are shown by way of illustration. It is to beunderstood that there are other embodiments and that structural andfunctional modifications may be made. Embodiments of the presentinvention may take physical form in certain parts and steps, examples ofwhich will be described in detail in the following description andillustrated in the accompanying drawings that form a part hereof.

Aspects of the present invention may be implemented with speciallyprogrammed computer devices and/or specially programmed computer systemsthat allow users to exchange trading information. An illustrativecomputing system specially programmed to implement a trading networkenvironment defining illustrative trading systems and methods is shownin FIG. 1.

An exchange computer system 100 receives orders and transmits marketdata related to orders and trades to users. Exchange computer system 100may be implemented with one or more specially programmed mainframe,desktop or other computers. A user database 102 includes informationidentifying traders and other users of exchange computer system 100.Data may include user names and passwords. An account data module 104may process account information that may be used during trades. A matchengine module 106 is included to match bid and offer prices. Matchengine module 106 may be implemented with software that executes one ormore algorithms for matching bids and offers. A trade database 108 maybe included to store information identifying trades and descriptions oftrades. In particular, a trade database may store informationidentifying the time that a trade took place and the contract price. Anorder book module 110 may be included to compute or otherwise determinecurrent bid and offer prices. A market data module 112 may be includedto collect market data and prepare the data for transmission to users. Arisk management module 134 may be included to compute and determine auser's risk utilization in relation to the user's defined riskthresholds. An order processing module 136 may be included to decomposedelta based and bulk order types for processing by order book module 110and match engine module 106.

The trading network environment shown in FIG. 1 includes computerdevices 114, 116, 118, 120 and 122. Each computer device includes acentral processor that controls the overall operation of the computerand a system bus that connects the central processor to one or moreconventional components, such as a network card or modem. Each computerdevice may also include a variety of interface units and drives forreading and writing data or files. Depending on the type of computerdevice, a user can interact with the computer with a keyboard, pointingdevice, microphone, pen device or other input device.

Computer device 114 is shown directly connected to exchange computersystem 100. Exchange computer system 100 and computer device 114 may beconnected via a T1 line, a common local area network (LAN) or othermechanism for connecting computer devices. Computer device 114 is shownconnected to a radio 132. The user of radio 132 may be a trader orexchange employee. The radio user may transmit orders or otherinformation to a user of computer device 114. The user of computerdevice 114 may then transmit the trade or other information to exchangecomputer system 100.

Computer devices 116 and 118 are coupled to a LAN 124. LAN 124 may haveone or more of the well-known LAN topologies and may use a variety ofdifferent protocols, such as Ethernet. Computers 116 and 118 maycommunicate with each other and other computers and devices connected toLAN 124. Computers and other devices may be connected to LAN 124 viatwisted pair wires, coaxial cable, fiber optics or other media.Alternatively, a wireless personal digital assistant device (PDA) 122may communicate with LAN 124 or the Internet 126 via radio waves. PDA122 may also communicate with exchange computer system 100 via aconventional wireless hub 128. As used herein, a PDA includes mobiletelephones and other wireless devices that communicate with a networkvia radio waves.

FIG. 1 also shows LAN 124 connected to the Internet 126. LAN 124 mayinclude a router to connect LAN 124 to the Internet 126. Computer device120 is shown connected directly to the Internet 126. The connection maybe via a modem, DSL line, satellite dish or any other device forconnecting a computer device to the Internet.

One or more market makers 130 may maintain a market by providingconstant bid and offer prices for a derivative or security to exchangecomputer system 100. Exchange computer system 100 may also exchangeinformation with other trade engines, such as trade engine 138. Oneskilled in the art will appreciate that numerous additional computersand systems may be coupled to exchange computer system 100. Suchcomputers and systems may include clearing, regulatory and fee systems.

The operations of computer devices and systems shown in FIG. 1 may becontrolled by computer-executable instructions stored oncomputer-readable medium. For example, computer device 116 may includecomputer-executable instructions for receiving order information from auser and transmitting that order information to exchange computer system100. In another example, computer device 118 may includecomputer-executable instructions for receiving market data from exchangecomputer system 100 and displaying that information to a user.

Of course, numerous additional servers, computers, handheld devices,personal digital assistants, telephones and other devices may also beconnected to exchange computer system 100. Moreover, one skilled in theart will appreciate that the topology shown in FIG. 1 is merely anexample and that the components shown in FIG. 1 may be connected bynumerous alternative topologies.

In an embodiment, a clearinghouse computer or computer system may beincluded. A clearinghouse or other entity that clears trades may use aspecially programmed clearinghouse computer or computer system forprocessing data to model and determine margin requirements.

Illustrative Embodiments

Clearing firms (e.g., a clearinghouse) may offer clearing services forone or more trading products, such as for futures. As part of theclearing services, the clearing firms may calculate margin requirementby relying on a risk management model that conforms to regulatoryrequirements and to the risk appetite of the particular clearinghouse.As such, a computer-implemented risk model should, therefore providegood coverage across a representative set of portfolios under acomprehensive set of historical scenarios, take into account all of thesignificant risk factors relevant to a plurality of futures products,consistently and proportionately model the effect of relevant riskfactors on the total risk exposure of portfolios including futuresproducts, and have robust, intuitive and justifiable parameterizationthat supports a reliable and transparent calibration and replicationprocess.

In some cases, a clearinghouse may rely on one or more models (e.g., ascenario-based model, a parametric model, a historical data model, etc.)to be implemented on an clearing computing system to determine marginingrequirements associated with customer portfolios. In an illustrativeexample, a computing system may implement a rule-based margining system,where margining may be first performed on synthetic portfolios of basictrading strategies that may include one or more of Outright, CalendarSpread, and Butterfly. In some cases, one or more scenarios (e.g., 16symmetric long and short scenarios) may be processed and then applied asmargin calculations. In such systems, the clearing computing system maydecompose each real portfolio into multiple synthetic portfolios wherethe final margin may be computed by aggregating the sub-marginstogether. In such cases, an end user will have to make the rules fordecomposition and setting the order for spread margin consumption,resulting in a very manually and computationally intensive marginingprocess because the methodology is rule based, where the rules can bedifferent for each financial product. Further, the clearinghousecomputing systems have faced increased strain and loading on thecomputational resources due to a rapid growth in new financial productsavailable to trade, where every real portfolio may include anycombination of these financial products. Each portfolio may bedecomposed into multiple synthetic portfolios, where a final margin maybe computed by aggregating the sub-margins calculated for each syntheticportfolio.

Inconsistencies may be introduced as the dimension reduction mechanism(e.g., tiers of financial products) is not consistent. Further,methodology of determining the tiers of financial products (e.g.,grouping contracts in terms of maturity), may not be statisticallysound. As such, margining of portfolios based on different strategies(e.g., Outrights, Calendar Spreads, Butterflies, etc.) for differenttiers is usually inconsistent. Further, only a limited number of spreadsmay be able to be margined where the rule-based margining may be limitedto particular portfolios of products. Further, such margining processesmay have limited scalability, without incorporating excessive computingpower and data storage capabilities, because the number of inter-curvespreads and the order in which they are applied may be limited. Further,this type of margining may have limited accuracy due to a reliance onrecent data points. Such a margining model may be used for a smallnumber of simple (e.g., “vanilla”, linear, etc.) futures products.However, such rule-based models may cause excessive computationalstrains to the clearinghouse computing system due to an exponentiallygrowing number of scenarios to be processed with or without offsettingrules and their order of application when calculating a marginrequirement for large number of financial products. Further, thecomputer systems processing the rule-based margining models may processseasonal futures products on an ad-hoc basis and/or may need increasedcomputational power to process margin requirements for multiple standard(e.g., vanilla) options products using one or more volatility surfacereconciliation methods (e.g., a price-volatility matrix, etc.). In somecases, the rule-based models may not be suitable for determiningmargining requirements for one or more exotic options products, such asan average price option (APO), a calendar spread option (CSO), a “strip”option, and the like, without significant computational and modeladjustments.

Parametric margining models may be useful for determining marginingrequirements for a group of homogeneous products or products having astable correlation structure. However, the parametric models may not beused by clearinghouse computing systems to efficiently calculatemargining requirements for a group of products having substantiallydifferent underlying risk factors (e.g., futures, implied volatilities,etc.) and/or for products having an unstable correlation structure. Theparametric model may introduce potentially difficult calibration issues,difficult choices in using a particular parametric model such as anoutright margin model, a correlation-based margin model and, differentdependence structures between product types (e.g., futures, options,etc.).

In some cases, the clearinghouse computing systems may processcomputer-executable instructions to utilize a historical data-basedmargining model that may be useful in determining margin requirementsfor a heterogeneous group of financial products in a straightforwardmanner using one or more historical scenarios. Such models may be dataintensive and dependent on the quality and amount of historical dataavailable and may be difficult to use for products with low dataquality, or having limited or no data history.

FIG. 2 shows a block diagram representation of a margin computationsystem 200 including a clearinghouse computing system 210 forcalculating a margin requirement according to aspects of the disclosure.The clearinghouse computing system 210 may include one or morecomponents (e.g., computing devices, modules running on one or morecomputing devices, etc.) that, when processing computer-executableinstructions, calculate an initial margin requirement for a portfoliocontaining one or more financial products (e.g., futures products, etc.)by: dimension reduction, variance scaling, covariance scaling, andValue-at-Risk (VaR) estimation. In some cases, the margin computationsystem 200 may include the clearinghouse computing system 210communicatively coupled via a network 205 (e.g., a wide area network(WAN), the LAN 124, the Internet 126, etc.) to a financial marketcomputing system 250. The financial market computing system 250 mayinclude one or more computing devices configured for receiving anddisseminating information corresponding to a financial market (e.g., afutures market, an options market, etc.), such as pricing information(e.g., bid information, ask information, etc.) and/or the like. Theclearinghouse computing system 210 may be communicatively coupled to ahistorical financial information database 260 via the network 205. Insome cases, at least a portion of the historical financial informationdatabase 260 may be otherwise incorporated into the clearinghousecomputing system 210 and/or the financial market computing system 250.

In some cases, the financial market computing system 250 may include adata repository, one or more computing devices and/or a user interface.The data repository may store instructions, that when executed by theone or more computing devices, may cause the financial market computingsystem 250 to perform operations associated with monitoring financialtransactions, receiving buy and/or sell orders, communicating financialinformation corresponding to pricing information for a plurality offinancial products offered via the associated financial market(s). Insome cases, the financial market computing system 250 may communicatefinancial information corresponding to the plurality of financialproducts in near real-time, at predetermined time intervals (e.g., about5 seconds, about 30 seconds, about 1 minute, about 2 minutes, at the endof a trading day, at the start of a trading day, etc.) determiningperformance bond contributions associated with holdings in products thatare based on various types of futures. In some cases, the financialmarket computing system 250 may store the financial informationcorresponding to the plurality of financial products in the historicalfinancial information database 260. Further, the financial marketcomputing system may further present the financial informationcorresponding to the plurality of financial products via one or moreuser interface screens via the user interface, whether the userinterface is local to the financial market computing system 250. Theuser interface may be local to the financial market computing system 250and/or remote from financial market computing system 250 and accessiblevia the network 205. The user interface screens may graphically and/ortextually present financial information corresponding to the pluralityof financial products in near real time, at predefined intervals and/orin response to a user request.

In some cases, the choice, calibration, and/or calculation of the riskrequirement employs a detailed statistical analysis of the risk factorsunderlying futures instruments held in the portfolio. In an illustrativeexample, the clearinghouse computing system 210 may include one or moreprocessors 212, one or more non-transitory memory devices 214 (e.g.,RAM, ROM, a disk drive, a flash drive, a redundant array of independentdisks (RAID) server, and/or other such device etc.), a user interface216 (e.g., a display device, a keyboard, a pointing device, a speaker, amicrophone, etc.), a data repository 218, a communication interface tofacilitate communications via the network 205, and/or the like. In somecases, the clearinghouse computing device 210 may be configured to storeinstructions in the one or more memory devices 214 and/or the datarepository 218 that, when executed by the processor 212, may configurethe clearinghouse computing device 210 to execute a model fordetermining margining requirements associated with a portfolio offinancial products, such as futures products, options products, stocks,and/or the like. In some cases, the clearinghouse computing system 210may process the instructions stored in the memory device 214 and/or thedata repository 218 to calculate the margining requirements using one ormore of a time series generator 230, a dimension reduction module 232, avariance scaling module 234, a covariance scaling module 236, avalue-at-risk (VAR) estimation module 238 and/or a margin calculator240. For example the clearinghouse computing system 210 may utilize thetime series generator 230, the dimension reduction module 232, thevariance scaling module 234, the covariance scaling module 236, thevalue-at-risk (VAR) estimation module 238 and/or the margin calculator240 to determine margin requirements for a portfolio of financialproducts based on one or more financial models, such as a risk model. Insome cases, the risk model may transform daily log returns of futureprices to orthogonal principal component (PC) scores and uses those PCscores as risk factors. The transformation is performed on rollinggeneric contract time series, which is a synthetic series constructed bygrouping existing contracts according to time-to-maturity. A roll-aheadfor generating generic contracts can be applied in order to providetime-series without any significant short term maturity effects.

In an illustrative example, the time series generator 230 may be used toprocess financial information (e.g., pricing information) correspondingto one or more financial products, such as financial products held in aclient's portfolio. For example, the time series generator 230 may beused to identify which financial products may be held in a clientportfolio for which a margin requirement is to be calculated. Forexample, the time series generator 230 may receive a customer identifier(e.g., an account number, a name, etc.) and retrieve account informationfrom a portfolio database 222 corresponding to the customer identifier.In some cases, the portfolio database 222 may store informationcorresponding to a plurality of customers, where the portfolios mayinclude information detailing a size of a holding of the differentfinancial products held in the portfolio and/or pricing informationcorresponding to the different financial products held in the portfolio.In some cases, the time series generator 230 may retrieve pricinginformation corresponding to the plurality of financial products held inthe customer's portfolio from a remote database (e.g., the historicalfinancial information database 260, etc.) via the network 205. Thispricing information may be used to process contract price seriesinformation to build time series of generic contracts (e.g.,representative financial products). In some cases, a rolling procedureis preferred over interpolation, an alternative approach to construct ageneric contract time series at fixed time to maturity. As such, thetime series generator 230 may process large amounts of historical datasets to generate a rolling generic time series. A core advantage of therolling approach is to allow a one-to-one mapping with the existingassociated financial product contracts (e.g., futures product). Theconcern is that rolling procedure may bring a periodic non-stationarity,which would be the strongest for the front-month generic contract.Empirical statistical analysis has shown that the time to maturityeffect in price daily log return distribution for the front monthgeneric contract is small. However, in the back-end of the termstructure curve, an interpolation or extrapolation technique may be usedto fill in or extend any missing data points along term structure inorder to keep the continuity of the rolling generic times series. Forexample, the time series generator 230 may process instructions toutilize linear extrapolation at the price level. In some cases, such aswhen the data time series of a product is much shorter compared withother products, an extension model may be designed to generateartificial data along time dimension for that specific product. The timeseries generator 230 is configured to allow use of the maximum amount ofdata available, and not to truncate long-time-series because some of thecurves have limited time-range. For example, a Student's t-distributioninverse conditional expectation formula may be used.

TABLE 1 Methodologies for use in time series generation Method Pros ConsRolling Clean time series without any Difficult to obtain roll-aheadshort term maturity effects parameter using indicator Testing possibleto determine variables is difficult due to impact of any short term theunreliability of the maturity effects curves Some returns are fullyuncovered by the models Interpolation Allows for different No historicaldata for interpolation methods based generics corresponding to on thedata set - linear contracts at the back-end of interpolation, cubicspline the curve interpolation, piecewise cubic hermite interpolation(PCHIP) Time series Allows for building multi- Capturing dependenceextension curve models for products between illiquid products havingdata sets starting at and liquid products different time points requiresa sufficiently large data set

In some cases, the portfolio database 222 may include portfolioinformation corresponding to a plurality of financial products, such asfutures products. In many cases, multiple futures products may be heldin a particular portfolio, particularly in cases where a large number offinancial products are available to the investor. For example, FIG. 4shows an illustrative listing of futures products corresponding to acorresponding commodity according to aspects of the disclosure. As such,data identification may be considered to be the backbone of the model,where information related to each of the multiple (e.g., about 100,about 500, about 1000, etc.) futures products may be processed toproduce a continuous time-series of information (e.g., a time series ofdaily settlement prices) for an each of the plurality of financialproducts. The time series of pricing information, such as settlementprices, may be generated by analyzing or otherwise processinginformation from a number of different contracts (e.g., about 50contracts, about 100 contracts, about 200 contracts, etc.) over a timeperiod (e.g., about 1 year, about 3 years, about 5 years, etc.). Thetime series information may reflect monthly contracts and/or may includeinformation corresponding to different market characteristics, such asseasonality, volatility related to a maturity date, and/or the like.

The time series generator 230 may process instructions stored in the oneor more memory devices 214 to generate one or more time series offutures products (e.g., a generic futures contract) according to one ormore approaches that may include a rolling generation process, aninterpolation generation process and/or the like. The time seriesgenerator 230 may retrieve pricing information via the network 205 fromthe historical financial information database 260 for a plurality offinancial products (e.g., futures products, etc.). As mentioned above,FIG. 4 shows an illustrative listing of futures products based ondifferent energy sources (e.g., commodities) such as crude oil, coal,natural gas, heating oil, gasoline, ICE Brent Crude, ICE Gasoil, ICE lowsulfur, Crude Gulf Sour, and the like. In some cases, time seriesinformation retrieved form the historical financial information database260 may be associated with starting dates for each raw time seriesassociated with a particular commodity, and/or a starting date andending date for use in the analysis. In some cases, the starting date ofthe time series may be different than the starting date for use in theanalysis. In some cases, an analysis starting date may be chosen to bethe same for different futures products. Further, in some cases such asin cases where less information is available (e.g., a newer developedfutures product), the start date of the time series may be the same asthe start date used in the analysis. Due to the varying length of thetime series information, the modeling of the information to be processedby the dimension reduction module 232, the length of the curves may bebased on a time interval between the start date of the analysis and theend data of the analysis. In some cases, the end date of the analysismay be a date earlier than a present date. For products having lessinformation (e.g., a shorter information time frame), the time seriesinformation may be extensible.

TABLE 2 Methodologies for use in the dimension reduction model MethodPros Cons Orthogonal Components High reactivity Lack of interpretabilityfor Fast computation illiquid components Parametric Model Control of theParameter instability and forward shape autocorrelation of parameterchanges Fitting issues for the front month Benchmark model Directmodeling No dimension reduction - of the generics not possible to use ona multi-curve level

In some cases, the time series generator 230 may process instructionscorresponding to a mathematical equation to determine a rolling timeseries for a plurality of futures products, such as in cases for“regular” futures products. In such cases, the time series generator 230may find the return time series R _(n) ^((g)) (t_(i)) of the n-thgeneric contract of a future product by the rolling procedure using:

$\begin{matrix}{{{R_{n}^{(g)}\left( t_{i} \right)} = {R_{T_{j + n - 1}}\left( t_{i} \right)}},{{{where}\mspace{14mu} t_{i}} \in \left( {{T_{j - 1} - \delta_{j - 1}},{T_{j} - \delta_{j}}} \right\rbrack},} & (1)\end{matrix}$

-   -   where Tj is the maturity date of contract with number j=1, . . .        , Ncnt and Ncnt is the total number of contracts. The initial        maturity date T₀ is defined as

$\begin{matrix}{T_{0} = {{Min}_{T}\left( {{T - \delta} > t_{0}} \right)}} & (2)\end{matrix}$

-   -   where T varies in the set of all available contract maturities        and to is the starting date of the analysis, as shown in FIG. 4.        Additionally, R_(Tj)(t) is given as the log-return at time t_(i)        of the traded futures with contract number j, as shown in:

$\begin{matrix}{{{R_{T_{j}}\left( t_{i} \right)} = {{\log \left( {F_{T_{i}}\left( t_{i} \right)} \right)} - {\log \left( {F_{T_{j}}\left( t_{i - 1} \right)} \right)}}},} & (3)\end{matrix}$

-   -   where F_(T)(t) is the price of the future contract with maturity        T at time t. For generality the roll-ahead period δ for the        rolling can be chosen as being maturity and product dependent,        however this roll-ahead period may be chosen as a fixed,        maturity and product independent constant due to the        unreliability of some data curves and also to avoid distortion        of the inter-curve correlation.

By construction the rolling procedure puts together returns withdifferent time to maturity (e.g., less than a month difference), whichmay cause a periodic nonstationarity. This effect may be the strongestfor the front-month generic futures. As such, a possible remedy mayutilize a predefined roll-ahead period when performing the shift fromone contract to another before the end of the contract. By doing so,errors introduced by rolling on contracts with very shorttime-to-maturity may be minimized or eliminated. Heuristics based onopen interests and traded volume for determination of the roll-aheadperiod may be analyzed by matching the rolling procedure of ETFcontracts. We may focus on front-month returns for which thetime-to-maturity effects may be expected to be the strongest.Practically, we compare the two extreme sets of log-returns availablewithin the front month time series, namely: (1) the shorttime-to-maturity set of returns with time-to-maturity less than or equalto 2 days, and (2) the long time-to-maturity set of returns withtime-to-maturity between about 28 and 31 days.

Two different tests may be run to compare these two distributions. Forexample, the first two moments may be tested using a two-sampleKolmogorov-Smirnov (KS) test, where the null hypotheses is that shorttime-to-maturity and long time-to-maturity returns are sampled from thesame distribution. This null hypothesis may be rejected if the teststatistics yields a p-value smaller than 5%. In the second test, amulti-generic case may be investigated using the correlations of thefront month contract to the second month contract. As the correlationmodel, an exponentially weighted moving average (EWMA) with apre-defined decay parameter (e.g., about 0.9) may be used. As theobtained correlations are very high, a visual check, or other automatedgraphical analysis method, may be performed on the commodities to ensurethere is no strong time-to-maturity effects exist, or are at leastminimized. FIG. 6 shows an illustrative two-sided KS test result toanalyze time-to-maturity effects on a return distribution according toaspects of the disclosure. In the chart, the fields marked with an X(e.g., ICE G and ULS), show that a test has been passed (e.g., H₀ hasbeen rejected) and that the distributions are different. The sample sizecolumn indicates the smaller sample size between the two sets (e.g., ashort set and a long set). For example, FIG. 7 shows an illustrativechart showing the distribution of the two samples used in the two-sidedKS test for a particular commodity according to aspects of thedisclosure.

From this single-generic analysis (the first KS test), it is rejectedthat the two sets come from the same distributions only for two ICEproducts (Gasoil and Low Sulfur). For these two commodities, the shorttime-to-maturity distribution is characterized by 0-returns, asillustrated in FIG. 7. Inversely, the result of these tests means thatfor futures products associated with the other commodities there may beno significant time to maturity effect observable. FIG. 8 illustrates asimilar result for WTI Crude Oil. Here the p-value is one of thesmallest along with the ICE Brent (e.g., 15%)) among the set ofcommodities that are rejected in the KS test. The time-to-maturityeffect in the rolling procedure may be statistically small and maypredominate for two ICE futures (Gasoil and ULS). The main rational forthe time series generator 230 to introduce a roll-ahead parameter forthe rolling is to counteract effects resulting from the decayingcorrelations of the front month contract to the second month contract.

FIGS. 9 and 10 show illustrative log-return correlations of the frontmonth contract to the second month contract using an exponential decaymodel for different financial products (e.g., crude oil and natural gas)according to aspects of this disclosure. For example FIG. 9 correspondsto exponentially weighted correlations (e.g., λ=0.9) of front to secondmonth returns for a financial product associated with a first commodity(e.g., crude oil) and FIG. 10 corresponds to exponentially weightedcorrelations (e.g., λ=0.9) of front to second month returns for afinancial product associated with a second commodity (e.g., naturalgas). By analyzing (e.g., visual analysis, image analysis, patternanalysis, etc.) such log-return correlations, either as a set ofinformation or as a graphical plot, one or more patterns may beidentified. For example, a clear decay may be identified in correlationfor financial products having very short maturities for differentcommodities (e.g., ICE gasoil, ICE ULS, WTI crude oil, and heating oil).Further, in some cases such as for ICE gasoil, a clear correlation breakmay be identified, as shown in FIG. 11. In some cases, no apparentchange in correlations can be identified, such as in the cases for coal,ICE Brent, natural gas, gasoline and Gulf Coast sour.

In some cases, the time series generator 230 we may process heuristicarguments to determine a sensible roll-ahead period for use with thetraded volume and/or open interest crossing time. FIGS. 12 showsillustrative charts showing traded volume curves for 6 consecutivecontracts associated with a particular commodity (e.g., WTI crude oil)and a distribution of crossing days for the front and second monthscontracts according to aspects of the disclosure. FIG. 13 showsillustrative charts showing open interest curves for 6 consecutivecontracts associated with the same particular commodity (e.g., WTI crudeoil) and a distribution of crossing days for the front and second monthcontracts according to aspects of the disclosure. In the illustrativeexample, FIGS. 12 and 13 illustrate the crossing days based on,respectively, the volume and open interests for the WTI crude oil. Here,the crossing time t_(c) may be defined as the first day for which theindicator (e.g., volume, open interest, etc.) of the next contracts islarger than the current contract's indicator:

$\begin{matrix}{{{t_{c}(j)} = {{{\min\limits_{t \leq T_{j}}t}{X_{j}(t)}} = {{X_{j + 1}(t)} < 0}}},} & (4)\end{matrix}$

-   -   where t_(c)(j) is the crossing time for which the current j        contract gets rolled with the next contract, and X denotes        either the open interest or traded volume. Further, the        parameter δ is determined by taking the difference with the        expiration date of the current contract:

δ(j)=T _(j) −t _(c)(j)   (5)

-   -   The parameter may be expressed directly in business days. If the        open interest or the traded volume does not cross during the        considered month, i.e. the curve of the current month is below        (or above) the curve of the next month, the value is not derived        and is reported separately. A set of indicators on this        measurements are displayed in FIG. 14 for the traded volume and        in FIG. 15 for the open interest. In some cases, an analysis of        one or more financial products (e.g., crude gulf sour (MB)) may        not be possible due to minimal trading volume information        available in the historical financial information database 260.        Nobs: number of observations. It corresponds to the total number        of curve intersection

In some cases, the content for of the tables of FIGS. 14 and 15 may bedefined as a number of observations (N_(obs)), that may correspond tothe total number of curve intersections considered, (e.g., for whichfront month data was available). In some cases, N_(obs) may includecases for which no crossing is observed on the front month. The numberof observations for which a crossing is observed may be denoted asN_(δ). Median, mean and standard deviation may be derived directly fromthe extract of the parameter δ for all the N_(δ) intersections. In casesfor which there are no crossings in the considered month, thatparticular month is not considered in any calculation of the median,mean or standard deviation. In some cases, the median, mean and/orstandard deviation may be expressed in business days, thus limiting themaximum reported value to 21 days for monthly calculations. In somecases, coverage may be defined as the ratio of the number ofobservations for which a crossing could be measured by the total numberof observations:

-   -   coverage=N_(δ/)N_(obs) (6) and may be expressed as a percentage.        In some cases, the mean number of intersections may indicate a        number of crossings of the front and second month's curves are        actually observed. For smooth curves like the ones shown in FIG.        13, only one crossing is observed, while for the traded volume        curves more crossing may be identified, as shown in FIG. 12.        N_(below) corresponds to the “front month below” and may be        defined as a number of curves for which the front month open        interest or traded volume was always below the second month for        any particular interval being analyzed. Similarly N_(above)        corresponds to the “front month above” and may be defined as a        number of curves for which the front month open interest or        traded volume was always above the second month for any        particular interval being analyzed.

From an analysis on all commodities, the traded volume may become highlyvolatile over the last days, making the measurement of the last crossingday difficult. For example, there can be more than one crossing of thecurves during this time period. Conversely, the open interest curves maybe more stable and their crossings may be easily predictable. In bothcases, the distributions of the crossing days are right skewed with insome cases where no crossings may be observed for a full month. Forexample, this may be the case for illiquid products (e.g., coal) or whenthere is a strong seasonal pattern in the demand (natural gas, heatingoil, etc.). To be robust with regards to this right skew the median ofthe distribution is proposed to retain only one roll-ahead by commodity.In some cases, several observations may be made based on an analysis ofthe tables and figures discussed above. However, in cases where minimaldata is available, such as for gulf coast sour, no derivation of valuesmay be possible. For example, the procedure can only be applied to thecrossing of the front month and the second month. For longer genericsmaturities the time series of open interest and traded volume are likelyto either not cross at all or to cross multiple times. Further, theseindicators cannot be used for futures with a low liquidity like coal,for which the curves usually do not cross at all. For commodities with astrong seasonal component, the crossing of open interest may becomplemented by a demand factor analysis to correct for structuralhigher or lower levels of supply, such as in a case concerning thestorage levels of natural gas. In some cases, the crossing of the openinterest curves may take place very early while the open interest of thefront month still remains very high, as determined in a case analysisfor ICE Brent as compared with WTI shown in FIG. 13 in which the openinterest of the front month is in clear decline when the two curvesmeet. Finally, the numbers may differ significantly between thedifferent liquid commodities, such as ICE Brent Crude and WTI Crude oil.

Having different roll-ahead parameters for different commodities wouldcreate major distortion of the inter-curve correlations. Moreover,obtaining the roll-ahead parameter using the indicator variables (e.g.,traded volume and open interest, etc.) is difficult due to theunreliability of the curves for one or more reasons, including multiplecrossing, no crossing at all in traded volume/open interest and/or thelike. For these reasons, we may calibrate the roll-ahead based on themedian crossing day of the open interest of crude oil (e.g., 7 days).Using roll-ahead for the generics has the advantage to provide a cleantime-series without any short term maturity effects. However it also hasthe disadvantage to leave a few returns fully uncovered by the models(the so-called generic 0). Final backtests may be performed with adecrease roll-ahead parameter to assess how large the impact of theshort time-to-maturity effect is once processed through all the steps.For instance, it could be captured and/or mitigated through theresiduals of the PCA approach by the dimension reduction module 232, andthus not impacting the multi-curve results.

In some cases, the time series generator 230 may process instructions toconstruct the returns of generic future contracts utilizes interpolationof price times series F_(Tj)(t). In some cases, the time seriesgenerator 230 may use one or more interpolation methods including linearinterpolation, cubic spline interpolation, piecewise cubic hermiteinterpolation (PCHIP). As a first step, the time series generator 230may retrieve historical information (e.g., pricing information)corresponding in maturity for a union of time points of the individualcontracts of the financial products (e.g., futures) in question. Next,the time series generator 230 may interpolate to daily granularity inmaturity. For example, for each time t, a data vector:

[F_(T) _(i) (t), . . . , F_(T) _(N) _((t))(t)]   (7)

may be used to determine an interpolating function F_(T)(t), where Tdenotes a refined (daily) grid in maturity dates. From the interpolatedforward curves (log) returns on the contract level may be computedusing:

r _(T)(t)=log(F _(t)(T))=log(F _(t−1)(T))   (8)

The prices and returns are transformed from maturities of contracts totime-to-maturity (ttm) by:

τ_(j) =T _(j) −t   (9)

where τ_(j) is the time-to-maturity of contract j. Without extrapolationat the front end of the curve the anchor points in the time-to-maturityare set as follows: starting from smallest time-to-maturity for allobservation dates (e.g., at the end of the front month) the time seriesgenerator locates an anchor point every 22 business days in maturitydirection. Given a set of points x_(i) and function values y_(i), withi=1. . . N, an interpolation method finds a smooth function graph f inbetween the grid points (x_(i); y_(i)). The function f(x) is called theinterpolating function for the data points (x_(i); y_(i)). Unless moreinformation on the desired function is known, such as an approximateanalytical form, the time series generator 230 uses simple functionalspaces to approximate the interpolating function. The most simple formis piecewise linear interpolation. The general form reads:

y=A|y _(j) +By _(j+1)   (10)

where the coefficients are given by:

$\begin{matrix}{A = {{\frac{x_{j + 1} - x}{x_{j + 1} - x_{j}}\mspace{14mu} {and}\mspace{14mu} B} = {{1 - A} = {\frac{x - x_{j}}{x_{j + 1} - x_{j}}.}}}} & (11)\end{matrix}$

The piecewise linear interpolating function has vanishing secondderivative and a discontinuous first derivative at the grid points.Because of the latter deficiencies a common alternative is interpolationusing cubic polynomials. Both, the second and third interpolationmethod, capitalize on this approach. Cubic splines are locally (cubic)polynomials that are glued together at the grid points by continuity andsmoothness constraints. The constraints include:

Interpolating function reproduces data points,

f _(i)(x_(i))=y _(i)   (12)

f _(i)(x_(i+1))=y _(i+1) , i=1 . . . N−1   (13)

first derivative matches at grid points, and

f′ _(i)(_(i+1))=f′_(i+1)(x _(i+1)), i=1 . . . N−2   (14)

second derivative matches at grid points.

f″ _(i)(x _(i+1))=f″ _(i+1)(x _(i+1)), i=1 . . . N−2   (15)

In particular, using cubic splines the interpolating function may berequired to generate a continuous first and second derivative. There are

4(N−1 )−2(N−1 )−2(N−2)=2   (16)

1leftover degrees of freedom that are fixed by boundary conditions. Weconsider the not-a-knot boundary conditions for the cubic splines.Therefore the three data points close to the boundaries (e.g., (y1; y2;y3) and (y_(N−2); y_(N−1); y_(N))) are each described by a cubicpolynomial. This leads to a matching condition for the third derivativeat the middle points:

f′″ ₁(x ₂)=f′″ ₂(x ₂), f′″ _(N−2)(x _(N−1))=f′″ _(N−1)(x _(N−1))   (17)

In comparison, PCHIP constrains the interpolating function in a firststep such that the function and its first derivative are continuous. Thetime series generator 230 may uniquely determine the interpolation basedon the data points y, and first derivative y′_(i) at the knots. Besidesthe conditions (12) the time series generator 230 has for PCHIP:

f′ _(i)(x _(i+1))=f′ _(i+1)(x _(i+1))=y′ _(i+1) , i=1, . . . , N−2  (18)

f′ ₁(x ₁)=y′₁   (19)

f′ _(N−1)(x _(N))=y′ _(N)   (20)

The time series generator 230 may process an interpolation routine toestimate the first derivative from the data numerically and chooses theslopes at the knots such that the shape of the data and the monotonicityis retained. However, using PCHIP may cause the second derivative of theinterpolant can be discontinuous. Generally, considering higher orderpolynomial interpolants is not useful as it leads to stronglyfluctuating interpolating functions. In an illustrative example ofliquid commodities (e.g. WTI crude oil), one or more of the discussedmethods may produce satisfactory results. As such, a quantitativecriterion for deciding which interpolation method would be mostpreferred may be used. Further, the stability and/or sensitivity of theinterpolation against missing data should be assessed. To do so, we maytest the stability of return distribution spreads between the firstand/or second data points by comparing:

r(t)=(F _(T) ₀ (t)−F _(T) ₁ (t))/F _(T) ₁ (t)   (21)

where 1; 2 are the ttm of the first two prices at fixed observation timet, to

r (t)=( F _(T) ₂ −F _(T) ₁ )/F _(T) ₁   (22)

where F_(τ2) is interpolated.

To process the test, we remove the second data point and repeat theinterpolation on the reduced data set. Using the 2-sampleKolmogorov-Smirnov (KS) with significance level α=0:05 the hypothesiswas tested as to whether the samples of spreads with genuine data andinterpolated data points are drawn from the same distribution. Resultingfrom this, in addition to other findings, is the rejection of linear andPCHIP interpolation for at least crude oil. In these tests, the full setof available contracts with a starting date of Jan. 3, 2001 were used.The FIGS. 17-19 show the corresponding histograms for the returndistributions all interpolation methods. For example, FIG. 17 shows a1-2 data point spread (e.g., return) distribution of crude oil withlinear interpolation, FIG. 18 shows 1-2 data point spread (e.g., return)distribution of crude oil with spline interpolation, and FIG. 19 shows1-2 data point spread (e.g., return) distribution of crude oil withPCHIP interpolation. The p-values of the KS test are given in FIG. 16 asrelated to equations (21) and (22). As expected cubic splines and PCHIPare perform similarly, where differences between the two splines areobserved especially where the removed point is a price maximum/minimum.Here, this result causes the rejection in the KS test for crude oil.Results for all commodities are presented in FIG. 20 which shows aresult table for various commodities according to aspects of thedisclosure. Here, fields that are shown to be marked (e.g., ‘X’) arerejected interpolation methods using the various KS tests and the blankfields are not rejected. The results for other illiquid commodities,like coal and golf coast sour, are inconclusive as all interpolationsare rejected (or not rejected) according to the KS tests.

FIG. 21 shows illustrative chart showing an example for the performanceof the spline interpolation and PCHIP for coal price data. Visualcontrol for specific dates favors PCHIP or linear interpolation becausecubic splines (with ‘not-a-knot’ boundary conditions) may introduce morevariance to the curve. The choice of interpolation method, however, doesalso depend on the settlement pricing of the considered products. Forexample, coal uses piecewise constant interpolation for settlementprices, which explains why cubic spline interpolation may not be asuitable method for coal futures.

For some commodities like coal the granularity of the maturity in theback-end of the futures curve decreases. Only specific contracts, suchas June or December contracts may be available. To keep the granularityin maturity constant when rolling generics returns across the back-endof the PFC, the missing prices are constructed using an interpolationprocedure. In some cases, the interpolation method that is best suitedto the corresponding commodity may be choses. For example, cubic splineinterpolation may be chosen for all but coal for which linearinterpolation may be favored. The returns of the filled futures curvemay be used to build the time series of generic contracts, such as bythe time series generator 230.

In an illustrative example, the time series generator 230 may identifythat in November 2013 the generic number 73 corresponds to the December2019 contract and that at day to that contract has to be rolled but thenext available contracts are June 2020, December 2020, June 2021, andDecember 2021. The 73rd generic return at time t₀+1 may be determined bythe time series generator 230 in the following steps:

-   1. interpolate the curves    -   [F_(Dec2013)(t₀), . . . , F_(Dec2019)(t₀), F_(Jan2020)(t₀),        F_(Dec2020)(t₀), F_(Jan2021)(t₀), F_(Dec2021)(t₀)]at the        maturity January 2020, obtaining F_(Jan2020)(t₀).-   2. interpolate the curve

[F_(Dec2014)(t₀+1), . . . , F_(Dec2019)(t₀+1), F_(Jan2020)(t₀+1),F_(Dec2020)(t₀+1), F_(Jan2021)(t₀+1), F_(Dec2021)(t₀+1)]  (21)

-   -   at the maturity January 2020, obtaining F_(Jan2020)(t₀+1)

-   3. The returns at time t₀ and t₀+1 for the generic 73 are    respectively

R ⁷³(t ₀)=log(F _(Dec2019)(t ₀))−log(F _(Dec2019)(t ₀−1))   (22)

R ⁷³(t ₀+1)=log(F _(Jan2020)(t ₀+1))−log(F _(Jan2020)(t ₀))   (23)

In some cases, the back end of the curve may occur when the number ofavailable contracts increases, particularly when little to no historicaldata may be available for the generics corresponding to these contracts.For in an illustrative example, new contracts for an entire yearmaturities may be listed in November and may correspond to the generics61-72 in December. These generics may not have a full historical timeseries, since the length of the futures curve oscillates between 61-72contracts. Reconstructing these data the curve length is extended to thefixed number of 72 by linear extrapolation of the available PFC curves.The generic returns may then be obtained from these extrapolated pricesas for real contracts. The same procedure may be applied when thehistorical curves are shorter in specific time periods than thecurrently observed time period.

In some cases, the time series of different basic products start atdifferent time points (see FIG. 4). For this reason we define anapproach that allows us to build multi-curve models in such a way thatlonger time series data (possibly including very important period ofcrises) may be efficiently exploited. An effective extension isespecially import for historical multivariate scaled innovations. Thoseinnovations are scarcely populated, especially in the tails, but may beheavily used in the historical simulation step.

In some cases, products having shorter time series may be handled indifferent ways depending on the liquidity of the particular products. Ifthe product is not liquid the shorter time series is problematic andtherefore may be modeled independently from the other curves. As aresult of the modeling, a margin for the problematic product M_(¬LP)with ¬LP representing a non-liquid product. This ensures that jointlymodeling the problematic curve with longer curves, does not relevantlydeteriorate the model performance of the longer curves of more liquidproducts. The total margins for a portfolio containing the problematiccurve among other products may then be obtained by adding the margin forthe illiquid product to the margins of the remaining portfolio.Therefore the total margin is given by:

$\begin{matrix}{M_{tot} = {M_{LP} + {M_{{LP}}.}}} & (24)\end{matrix}$

This approach may be justified by the fact that illiquidity leads toprice movements which are not observed in standard time series dynamicsof liquid products. Capturing the dependence structure between illiquidproducts and liquid ones a sufficiently large data set is required. Forilliquid products, with time series that does not satisfy the sufficientdata requirement, this attempt could lead to spurious correlationeffects. This, in turn, may decrease the needed margins in a unjustifiedway. Decoupling of the time series modeling and of the margincalculation is therefore a necessary step in the case of scarce data andlow data quality.

When we have a short time series for a liquid product, a considerationremains regarding a situation that a product is only very recentlytraded. In that case a meaningful inter-curve dependence structurecannot be estimated and the product may be treated as being analogous toilliquid products. In all other cases a joint modeling and marginestimation procedure with longer time series curves is desirable. Byassumption the length and the quality of the considered time seriesallow the estimation of the dependence structure of this curve withother products, on the period where all products are available. Inparticular, the dynamic of the co-variance matrix can be estimated andis used in filtering the multivariate returns. Furthermore, this impliesthe possibility to estimate higher order dependence effects which aregenerally assumed as static and left in the filtered innovations. Theextension of the curves is only done at the levels of innovations toavoid reconstructed and/or extended returns biasing the dynamic modelestimation.

For times series extension of scaled returns, N curves with returnsR_(i) ^((n)), where I labels the single futures on the curve and n maybe assumed with starting dates of time series S_(n) as ordered inascending order), n=1, . . . , N. At current time point t₀ we considerthe following set of time series

[R ⁽¹⁾(|t), . . . , R ^((n))(t)], t∈[S _(n) , t ₀], n=1, . . . , N  (25)

We refer to the FIG. 22 for further illustration of the introducednotation. FIG. 22 shows an illustrative representation of a time seriesextension according to aspects of the disclosure. For example, FIG. 22shows a representation of equation 25, where N=3 and the boxes indicatethe set of return time series for different time intervals. The samedimensional reduction and filtering procedure described below is appliedto the return time series to get the variance and co-variance scaledscores UI (so-called uncorrelated innovations)

[UI ^((1,n))(t), . . . , UI ^((n,n))(t)], t∈[S _(n) , t ₀], n=1, . . . ,N   (26)

where the second index in UI^((l,n))(t) describes which set of returnshas been used to estimate the correlation matrix. A multivariate studentt-distribution τ(μ, Σ, υ) is calibrated to the data:

[UI ^((l,N))(t), . . . , UI ^((N,N))(t)], t∈[S _(N) , t ₀]   (27)

using an expectation maximization (EM) procedure.

By using the EM algorithm the time series generator 230 can iterativelydetermine the maximum log-likelihood estimate for parameters of astochastic distribution. Here, the EM algorithm may be used to determinethe parameters of the multivariate t-distribution, (μ, Σ, υ). Taking theexample of FIG. 22, μ is a 3×1 mean vector, Σ is a 3×3 covariance matrixand u is a scalar for the degrees of freedom. This distribution may befitted on the period where all curve data are available, such as thelargest set of return time series equation. (27). This describes thestatic dependence structure of the scores innovations and can be used togenerate the missing data UI^((k,N))(t) with t ∈[S1, . . . , S_(k)−1](compare FIG. 22). To this end the following conditional distributionformula may be used:

$\begin{matrix}{{X_{1}{X_{2} \sim {\tau\left( {\mu_{12},{\sum\limits_{12}{,v_{12}}}} \right)}}},} & (28) \\{{\mu_{12} = {\mu_{1} + {\sum\limits_{12}{\sum\limits_{22}^{- 1}\left( {X_{2} - \mu_{2}} \right)}}}},} & (29) \\{{v_{12} = {v + {\dim \left( X_{2} \right)}}},} & (30) \\{{\sum\limits_{12}{= {\frac{v + {\left( {X_{2} - \mu_{2}} \right){\sum\limits_{22}^{- 1}\left( {X_{2} - \mu_{2}} \right)}}}{v_{12}}\left( {\sum\limits_{11}{- {\sum\limits_{12}{\sum\limits_{22}^{- 1}\sum\limits_{21}}}}} \right)}}},} & (31) \\{where} & \; \\{\left\lbrack {X_{1},X_{2}} \right\rbrack \sim {\tau \left( {\left\lbrack {\mu_{1},\mu_{2}} \right\rbrack,\begin{bmatrix}\sum\limits_{11} & \sum\limits_{12} \\\sum\limits_{21} & \sum\limits_{22}\end{bmatrix},v} \right)}^{-}} & (32)\end{matrix}$

holds.

In the present case we use (28) to generate simulated score innovation

X ₁=[UI ^(k)(t), . . . , UI ^(N)(t)]   (33)

for t ∈[S_(k−1), S_(k)−1], conditional on the available data

X ₂=[UI ^((l,k−1))(t), . . . , UI ^((k−1,k−1))(t)]   (34)

for all k=2, . . . , N. Note that for the conditional distribution theparameters

μ₁, μ₂, Σ₁₁, Σ₁₂, Σ₂₁, Σ₂₂, υ   (35)

are obtained directly from the parameters μ, Σ, υ of the multivariatestudent t-distribution that may be calibrated on [S_(N), t₀]. SettingUI^(j)(t)=UI^((j,k−1))(t) for t ∈[S_(k−1), S_(k)−1] and j≥k, results inthe extended scores innovations UI^(k)(t) for all curves k and the wholetime range t ∈[S₁; t₀]. In comparison, the error terms time series maybe extended simply by bootstrapping historical errors for each curveindependently. Note that X1 and X2 may be collections of curves and theEquations (27) to (34) may be in general for a multi-curve case.Regarding the example the FIG. 11 there are two steps in the extension:first, X₁=[UI³] for [S₂; S₃−1] under the condition X₂=[UI¹;UI²] andsecond, X₁=[UI²;UI³] is filled on [S₁; S₂−1] under the conditionX₂=[UI¹].

In an illustrative test case, the approach is tested on the volatilityscaled WTI and ICE Brent returns for the period from 1 Jan. 2010 to 29Nov. 2013. First we compute the volatility scaled return for the timeperiod 9 Aug. 2010 to 29 Nov. 2013 based on the original time serieswith a burn-in period. Afterwards we artificially cut the ICE Brentreturn time series to 1 Jan. 2011 to 29 Nov. 2013 and calculatevolatility scaled returns based on 9 Aug. 2011 to29 Nov. 2013. Next, thecut time series may be extended with the described approach to theperiod 9 Aug. 2010 to 9 Aug. 2011. We then compare real innovationsversus artificial innovations by overlapping the WTI-ICE Brent jointdistribution as in FIG. 23 and FIG. 24 shows observed versus extendedICE Brent innovations time series, while FIG. 25 presents a QQ plot ofthe innovation distribution. The real and extended time series of ICE Binnovations are given in FIG. 26. As shown, this approach, asimplemented by the time series generator, allows time series extensionsby roughly preserving the distributional properties of the consideredtime series. The time series, when created by the time series generator230 may be stored in the one or more the memory devices 214 and/or thedata repository 218

The dimension reduction module 232 may be communicatively coupled to thetime series generator 230 and may process instructions stored in the oneor more memory devices 214 to perform dimension reduction on the timeseries generated by the time series generator 230. In general, dimensionreduction refers to the process of capturing the curve dynamics in a fewfactors instead of all generics as may be required in the context of themulti-curve model, which uses a matrix Cholesky decomposition for thedynamic correlation model. Note that the dimension reduction techniquedoes not imply any loss of volatility, as the residuals (the omitteddimensions) are still considered in the single curve case. The onlyassumption made is that the correlation dynamic, at the multi-curvelevel, is described by a reduced set of factors. Three alternatives areinvestigated to decrease the dimensionality. For example, the orthogonalcomponent, i.e. the PCA method, as determined by the dimension reductionmodule may allow for high reactivity and fast computation but may lackinterpretability for illiquid commodities. A parametric model may allowthe dimension reduction module 232 to control the forward shape of thedimension reduced time series, however parameter instability and/orauto-correlation of parameter changes may be experienced along with fitissues associated with the front month. A Benchmark model which directlymodels the generics may be simple, however this model does not includeany dimension reduction and may be difficult, at best, for use on amulti-curve model.

The orthogonal curve model may be used for high correlations between thedifferent generic contracts to achieve a significant dimensionalreduction. Accounting for the short-term volatility dynamics, theorthogonal model is investigated for a covariance matrix C of thegeneric futures provided by an exponential weighting with decay μ_(PCA):

$\begin{matrix}{{{C_{i,j}(t)} = {{\lambda_{PCA}{C_{i,j}\left( {t - 1} \right)}} + {\left( {1 - \lambda_{PCA}} \right)\left( {{R_{i}(t)} - {{\langle R\rangle}(t)}} \right)\left( {{R_{j}(t)} - {{\langle R\rangle}(t)}} \right)}}},} & (36)\end{matrix}$

where R,(t) is the log-return at time t of the generic future i. TheEWMA estimate (R)(t) is derived as

$\begin{matrix}{{(R)(t)} = {\lambda_{PCA}\left( {{R\left( {t - 1} \right)} + {\left( {1 - \lambda_{PCA}} \right){R(t)}}} \right.}} & (37)\end{matrix}$

using the same λ_(PCA) parameter as in equation (36). The estimatedcovariance is diagonalized

$\begin{matrix}{{{C(t)} = {{{PC}(t)}{\Lambda (t)}{{PC}^{T}(t)}}},} & (38)\end{matrix}$

where Λ(t)=diag(λ(t), . . . , λ_(N)(t)) (in descending order) andPC=[PC₁(t), . . . , PC_(N)(t)] is an orthonormal matrix. The principlecomponent (PC) matrix defines a rotation and this transformation is usedto decompose the returns into uncorrelated components (“scores”). Usingthe PC matrix the kth score is defined as:

$\begin{matrix}{{{{pc}_{k}\left( {t^{\prime},t} \right)} = {\sum\limits_{j = 1}^{N}{{{PC}_{jk}(t)} \cdot {R_{j}\left( t^{\prime} \right)}}}},} & (39)\end{matrix}$

where t′ is the time indicator in the training window, while t is thederivation date of the PCs. Note that the scores can be rotated back, aswill be done at the generic simulation step by the dimension reductionmodule 232:

$\begin{matrix}{{R_{i}\left( {t^{\prime},t} \right)} = {\sum\limits_{k = 1}^{N}{{{PC}_{ik}(t)} \cdot {{{pc}_{k}\left( t^{\prime} \right)}.}}}} & (40)\end{matrix}$

The dependencies on both time t and t′ in the scores and returns areused to clearly track the time dependence of the PC matrix. A uniquetime dependence may be fixed to distinguish between updated andnon-updated scores. For updated scores the PC transformation of thederivation day may be used by the dimension reduction module 232 to findthe time series of the scores. For the non-updated scores thetransformation matrix PC may be time-aligned with the time series of thescores. Note that for the transformation of the scores (and residuals)back to return space the most recent PCs are used in both, the updateand non-update case. Considering the high correlations between thedifferent contracts, we consider only the first n components explainingthe majority of the dynamics. The share of explanation for each PC maybe provided directly by the ratio of the individual eigenvalue Λ(t)_(kk)to the sum of the eigenvalues.

The returns R(t′) can then be re-expressed as

$\begin{matrix}{{{R_{i}\left( {t^{\prime},t} \right)} = {{\sum\limits_{k = 1}^{N}{{{PC}_{ik}(t)} \cdot {{pc}_{k}\left( t^{\prime} \right)}}} + {ɛ_{i}\left( t^{\prime} \right)}}},} & (41) \\{{ɛ_{i}\left( {t^{\prime},t} \right)} = {\sum\limits_{k = {n + 1}}^{N}{{{{PC}_{ik}(t)} \cdot {pc}_{k}}{\left( t^{\prime} \right).}}}} & (42)\end{matrix}$

The quantities ε_(i) are commonly known as compression error. This termis misleading in this context as these contributions are not dismissedin the returns. As later explained, distinguishing scores and errors areonly important on the simulation level.

FIG. 27 shows an illustrative chart showing explained variance per PCfor crude oil according aspects of the disclosure. From a visual check,the explanation power of the different PCs is not distinguishable as ofthe 5^(th) PC. FIG. 28 shows an illustrative chart showing explainedvariance for a 99% threshold selection criteria for a financial productusing a covariance matrix estimated with a lambda of 0.98. The jumpscorrespond to the switch from 2 to 3 PCs. FIG. 29 shows an illustrativechart showing a number of selected PCs needed to reach the 99% varianceexplained threshold for crude oil and FIG. 30 shows explained varianceusing constantly three factors for crude oil.

The dimension reduction module 232 may determine an appropriate numberof scores according to one or more selection processes. For example, avariance based selection may be used or a constant number of factors maybe used. For a variance based selection, the number n of PCs selected isdefined as the minimum number of scores needed to explain a givenpercentage of variance, usually 99%. The advantage of this approach isthe direct control of the risk explained by the factors, relatively tothe error term. The disadvantage is that the number of PCs may change inthe course of time, thus creating discontinuity in the multi-curvecorrelation model for the non-updated case. A practical test case isprovided in FIGS. 28 and 29, where the achieved level of varianceexplanation and the number of PCs selected are reported for crude oil. Amaximum of 3 PCs are needed to explain this level of variance. Thenumber of selected PCs fluctuates between 2 and 3 as of 2011. Thisresult is reflected in FIG. 16, where the first two PCs sum up to 97%.

In some cases, the dimension reduction module 232 may receive a numberof factors n from a user via a user interface, where the number offactors are fixed by the user. A usual estimate is 3. The estimateshould be high enough to explain at least around 99% of the variance.The advantage of this approach is not only having a threshold level ofexplained risk, but also controlling the number of factors for themulti-curve model.

An example is displayed in FIG. 30 for crude oil. As FIG. 30 indicatesthree factors are always enough to explain at least 99% of the curve forWTI crude oil. The proposed parameterization is to use a constant numberof PCs, which would be monitored by commodities. If the share ofexplained variance falls below a threshold, the number of componentswould be increased. As a result one avoids a fluctuating number of PCsaround an arbitrary value (in this example, 99%).

FIG. 31 represents the time-series of the leading three scores of crudeoil using updated PCA. From a visual check, this time series exhibitsvolatility clustering and heavy tails, as does any financial timeseries. Since the PC vectors are orthonormal, the scale of the scoretime series gives a direct indication of each score's contribution tothe total variance. FIGS. 33 to 34 illustrate and quantify this dynamicsfor crude oil. For example, FIG. 33 shows an autocorrelation of thefirst scores, in an updated setting, from a PCA on crude oil. FIG. 34shows an autocorrelation of the first scores squared (in the updatedsetting) from a PCA on crude oil. The negative autocorrelation on thefirst PC may be translated into the mean-reversion feature of commodityprices, while the strong positive auto-correlation of the squared scoresquantifies the volatility clustering. The volatility model chosen in thenext section will require capturing this effect. FIG. 32 shows anillustrative table corresponding to visual inspection of PC surfacestability for the different curves according to aspects of theinvention.

In some cases, the dimension reduction module 232 may identify a stableset of time series. For example, a core assumption behind the orthogonalcomponent approach is that the PCs exhibit a certain stabilitycharacteristic. This feature is an important factor to have stableparameters of the multi-curve model, and for monitoring the updateprocess of the scores volatility. Beyond that, this feature could proveuseful in the context of injecting stress and risk views in the model.As a stylized fact, the first three PCs of a PCA on a forward surface(whether commodities or interest rates) usually yield respectively thelevel, the slope and the curvature. The first three PCs of WTI crude oilare represented by FIGS. 35A-C. Visually, the level, slope and curvatureappear clearly. Importantly, the small decay in the level componentillustrates a higher volatility in the front-end of the curve and isconsistent with the Samuelson effect. FIGS. 35A-C show a visualrepresentation of an inspection of PC surface stability for a pluralityof curves according to aspects of the disclosure.

The dimension reduction module 232 may identify one or more orthogonalcomponents associated with a time series. The orthogonal components arerotating the generic time series in an orthonormal space according toinitial exponential decay used for the covariance matrix. The scoresthus do not have a vanishing correlation between one another. Twoapproaches may be used by the dimension reduction module 232 to tacklethis issue.

The first approach may involve updating of the scores at everytime-step, the full history of the scores is derived based on the latestorthonormal basis. The modified version of Eq. (39) therefore reads:

$\begin{matrix}{{{{pc}_{k}\left( t^{\prime} \right)} = {\sum\limits_{j = 1}^{N}{{{PC}_{ik}(t)} \cdot {R_{j}\left( t^{\prime} \right)}}}},} & (43)\end{matrix}$

where t is fixed to the end of the time series, i.e. t′<t. The updatesetting of the PC analysis is favored when the correlation structure isstrongly changing over time and the assumption of stable orthogonalcomponents is not justified. For the first two scores pc₁(t′) andpc₂(t′) the dynamic correlation may be estimated using equation (36)with λ=0.97. We normalize the covariance following

$\begin{matrix}{\rho_{12} = \frac{{Cov}\left( {{pc}_{1},{pc}_{2}} \right)}{\sqrt{{Cov}\left( {{pc}_{1},{pc}_{1}} \right)}\sqrt{{Cov}\left( {{pc}_{2},{pc}_{2}} \right)}}} & (44)\end{matrix}$

to obtain the dynamical correlation. The derivation date of the PCsbetween may vary between 2008 and 2013. For time-steps far in historywith regards to the derivation date, the local dynamic correlation ofequation (44) can be very different from zero. FIG. 36 illustrates thiseffect. The scores, rotated with the covariance matrix derived inNovember 2013, get a strong negative correlation in 2007. Theinterpretation of this negative correlation with regards to the PCs ofNovember 2013 is complex, and can combine for instance a differentintensity of the Samuelson effect with a different dynamic of the secondcomponent (inversion of the curve slope). FIG. 36 shows an illustrativechart showing dynamic correlation of time series scores associated witha financial product according to aspects of the disclosure.

In some cases, the dimension reduction module 232 may be configured notto update any scores for the time series. The alternative to the fullupdate of the history of the scores is to keep the historicalprojections in time. Practically, Equation (39) becomes:

$\begin{matrix}{{{pc}_{k}\left( t^{\prime} \right)} = {\sum\limits_{j = 1}^{N}{{{PC}_{jk}\left( t^{\prime} \right)} \cdot {{R_{j}\left( t^{\prime} \right)}.}}}} & (45)\end{matrix}$

The historical transformation vectors PC_(jk)(t′) are used instead ofthe derivation date vectors PC_(jk)(t). The resulting correlations(computed using Equation (44)) may fluctuate between about 50% and about−50%, with peaks around the financial crisis, indicating that locallythe correlation structure changed significantly. For example, FIG. 37shows a dynamic correlation of the first and second scores in theno-update case for crude oil using a covariance matrix derived with alambda 0.98. Further, when contrasted with FIG. 35 there is no need torepresent different derivation dates as they would lead to a same curve.

The basic assumption behind the no-update case is that the correlationbetween the scores is small enough to allow to be considered asuncorrelated. Even in regimes showing dynamical correlation the noupdate procedure can be advantageous since it results in lessfluctuating scores. The gained stability of the scores can possibly leadto more stable (and satisfactory in terms of VaR coverage andclustering) margins as compared to the update procedure. Note that ifthe correlation structure is very stable, i.e. the dynamical correlationbetween scores is small, the update and non-update procedure lead tosimilar results for the scores. In this case the latter procedure may beused for computational reasons. The advantage of the no-update procedureis that only the last point is added to an existing set of scores, thusmaking the calculation extremely fast and providing extra control on theupdate process of the risk measures. A drawback of the no-updateapproach is that it assumes a high stability of the PCs. In particular,the PCs should not flip or fluctuate strongly, which brings somelimitations in the context of curves with a lesser liquidity.

There are no strong theoretical arguments to reject or prefer the updateor non-updated approach. The decision may be made by evaluating VaR backtesting results. Performing a PC analysis we are selecting NPC from theN components pc_(k)(t); k=1, . . . ; N in equation (39). Thesecomponents explain almost all the observed correlation. The remainingN_(Generics)-N_(PC) components are the residuals. Although they explainless than 1% of the price surface variance, they still contain somevolatility dynamics and so should not be considered as pure white noise.FIG. 39 represents this set of time series for an orthogonal model onWTI crude oil. The scale of the time series (peaks around 0.01) may becompared with the scale of the scores in other examples. In some cases,only the third score has a similar scale. As can be seen in FIGS. 38 and39 for crude oil some volatility clustering is contained in theresiduals, and thus a dynamic volatility model is useful also for theresiduals.

Dimension reduction can also be achieved using a parameter-dependentmodel. As a result, the future prices are described by the timeevolution of the parameters. We have investigated two models two-factormodel where the price FT (t) given by:

-   -   two-factor model of Schwartz/Smith [2] where the price        F_(T)(t)|is given by

$\begin{matrix}{{{\ln \; {F_{\tau}(t)}} = {X_{t}^{1} + {X_{t}^{2}\tau} + {X_{t}^{4}\left( {1 - \epsilon^{{- X_{t}^{2}}\tau}} \right)} + {\left( {1 - \epsilon^{{- 2}X_{i}^{2}\tau}} \right)X_{i}^{5}}}},} & (46)\end{matrix}$

-   -   -   where X₁ ^(i) with i=1 . . . 5 are parameters and τ is the            time to maturity.

    -   polynomial fitting model

$\begin{matrix}{{F_{\tau}(i)} = {{X_{t}^{5}\tau^{4}} + {X_{t}^{4}\tau^{3}} + {X_{t}^{3}\tau^{2}} + {X_{t}^{2}\tau} + X_{t}^{1}}} & (47)\end{matrix}$

-   -   -   where X₁ ^(i) with i=1 . . . 5 are the model parameters.

Following the fit of the models to crude oil data we investigate thestability of the parameter time series, test for autocorrelation in theparameters differences and apply AR-GARCH filtering in order to removevolatility clustering and autocorrelation effects. Both models arefitted to and tested on the data set June 2006-December 2013. Theparameters of the Schwarz/Smith model are estimated using non-linearoptimization of the squared error:

$\begin{matrix}{\mspace{79mu} {{s = \left. {\sum\limits_{i}\underset{\text{?}}{\left( {{\overset{\sim}{F}}_{i} - F_{i}} \right)^{2}}}\rightarrow\min \right.},}} & (48) \\{\text{?}\text{indicates text missing or illegible when filed}} & \;\end{matrix}$

where i indexes all available price data and ˜Fi are the modeled prices.Gauging the quality of the model description on crude oil we observeoverall satisfying model description except at the front end of thefutures curve.

The FIGS. 40 and 41 show the model residuals for crude oil with andwithout regularization. FIG. 40 shows a residual surface for crude oilwith the 2 factor model (46) without regularization. FIG. 41 shows aresidual surface for crude oil with the 2 factor model (46) withregularization.

The time evolution of the first parameter X_(t) ^(i) is depicted in FIG.42. FIG. 42 shows an illustrative chart showing a time evolution of thesecond parameter of the 2 factor model without regularizations. FIG. 43shows an illustrative chart showing a time evolution of the secondparameter of the 2-factor model using the regularization. The individualparameters show a rather erratic time dependence which is hard to treatwith time series modeling techniques. To avoid large jumps in theparameter evolution we change the objective function (48) by anadditional regularization term. The regularized objective function readsof the form

$\begin{matrix}{\; {{s = {{\sum\limits_{i}\left( {{\overset{\sim}{F}}_{i} - F_{i}} \right)^{2}} + {\lambda {\sum\limits_{j}\left( {X_{i}^{j} - X_{i - 1}^{j}} \right)^{2}}}}},}} & (49)\end{matrix}$

where j indexes the parameters. The results for the time series of thefirst parameter X1 with regularization term, for λ=1, is shown in FIG.44. Additional time series for the parameters can be found in AppendixC. As the next step the autocorrelation of the time series of parameterdifferences

ΔX _(t) ^(i) =X _(t) ^(i) −X _(t−1) ^(i) , i=1, . . . , 5   (50)

is assessed and the (augmented) Dickey-Fuller test is used to test fornon-stationarity. The autocorrelation up to 30 lags for the firstparameter is given in the FIG. 44. We include the autocorrelation plotsof the remaining parameters. The null hypotheses of vanishingautocorrelation based on the Ljung-Box test is consistently rejected for1 parameter. As autocorrelations are present in the parameterdifferences (50) a AR(1)-GARCH(1,1) model for filtering. Following thecomputation of innovations (58) we check again for autocorrelation inthe innovations. The test result is shown for the first parameter in theFIG. 45. Clearly, the autocorrelation in the innovations is reduced andthese innovations are used for risk estimation and backtesting. Ourresults for the VaR violation backtesting show that the VaR estimationis biased towards the front end of the curve. FIG. 44 shows anillustrative chart showing autocorrelation for a 2-factor model system.FIG. 45 shows an illustrative chart showing autocorrelation usingregularization.

Note that the changes in the futures price may be modified by a firstorder approximation parameter changes

$\begin{matrix}{{{\Delta \; F} = {\sum\limits_{i}{{\nabla_{X_{i}}{F(X)}}\Delta \; X_{i}}}},} & (51)\end{matrix}$

where we for simplicity left out the time arguments in the future pricesF and the parameters X. The term ∇F(X) is the gradient on the futureprice surface with respect to the changes in the parameters X. For the2-Factor model:

$\begin{matrix}{\mspace{79mu} {\frac{\partial{F(X)}}{\partial X_{1}} = 1}} & (52) \\{\mspace{79mu} {\frac{\partial{F(X)}}{\partial X_{2}} = \tau}} & (53) \\{\mspace{79mu} {\frac{\partial{F(X)}}{\partial X_{3}} = {{X_{i}^{4}{\tau ɛ}^{{- X_{i}^{3}}\tau}} + {2X_{i}^{S}\tau \; ɛ^{{- 2}\; X\text{?}}}}}} & (54) \\{\mspace{79mu} {\frac{\partial{F(X)}}{\partial X_{4}} = {1 - ɛ^{{- X_{i}^{3}}\tau}}}} & (55) \\{\mspace{79mu} {\frac{\partial{F(X)}}{\partial X_{5}} = {1 - ɛ^{{- 2}X_{i}^{3}\tau}}}} & (56) \\{\text{?}\text{indicates text missing or illegible when filed}} & \;\end{matrix}$

Moreover, ΔX denotes the parameter changes in time, cf. Eq. (50). Due tothe non-linear dependence of the future price on the parameters,historically observed parameter changes ΔX can result in wronglyestimated price changes in the present. In formulae, one uses

$\begin{matrix}{{{\Delta \; F} = {\sum\limits_{i}{{\nabla_{X_{i}}{F\left( X_{0} \right)}}\Delta \; X_{i}}}},} & (57)\end{matrix}$

where X₀ denotes the present parameter values, to compute the pricesdistribution instead of mapping historical parameter changes tohistorical price changes using (51).

FIG. 46 shows an illustrative chart showing results for VaR99 violationcoverage test for the 2 Factor model for outright long (upper panel) andoutright short (lower panel positions on a financial product. FIG. 47shows an illustrative residual surface for crude oil with the polynomialmodel 47 and without regulation. FIG. 48 shows an illustrative chart ofa time series of the first parameter for the polynomial model withoutregularization.

In some cases, the dimension reduction module 232 may operate using apolynomial model according to aspects of the disclosure. The residualsare plotted in the FIG. 47 without regularization. We find again thatthe overall fit to data is satisfactory except for the front month. Theparameter dynamics (FIG. 48) are more stable compared to the 2-Factormodel and, from this perspective, there is no need to introduce aregularization. The autocorrelation in the parameter changes, shows aslow decay and indicates a complex model dynamics (with possibly longmemory). The VaR violation plots show no bias with respect to positiveor negative returns as it is expected from a linear model. Theperformance is however not satisfactory, in terms of VaR violation ratesand clustering, at the front end of the curve.

The approach using parametric models presents already of series ofdifficulties at the single curve level. Both models have problemsdescribing the front end of the curve. Additionally, future curves withhigh curvature are fitted rather poorly. Without including aregularization term in the fitting procedure the parameters is instable(for the 2-factor model). These instabilities are partly cured by thepenalization of large jumps in the parameter time series. However, theparameter changes are strongly autocorrelated. These autocorrelationscan be successfully reduced using AR filtering for the 2-Factor modelbut more investigations are needed for the polynomial model. Theautocorrelations in polynomial model do belong to the AR-GARCH(1,1)model type and the removal of such effects by filtration is lesssuccessful. The tests on VaR violation fail for the front end of curvewhere the model description for both functional forms is poor.Additionally, the VaR estimation shows biases for long and shortpositions for the 2-Factor model. The biases could be due to thenot-accounting for non-linear effects when mapping the simulations ofparameters to price simulations. Note that for the parametric models the“scores” are the parameter changes X. In detail, from the filteredsimulations of the parameter changes ΔX the forecast price distributionis computed using the linear relationship (51). The gradient ∇F(X) isestimated using the parameters at derivation date (or the date of theforecast)—not the dates with each date of the historical parameters.Given the increased complexity of a satisfactory parametric model evenbefore considering multi-curve settings, the PCA method to dimensionalreduction of the price future data are clearly favored.

The variance scaling module 234 may use any number of approaches tomodel volatility of the scores. Three approaches are investigated tomodel the volatility of the scores. Additionally, engineering techniquesare reviewed to constrain the pro-cyclicality feature of the riskmeasures, namely the volatility flooring and the dual lambda approach.The methods may include EWMA which allows fast computation and/orcontrol of the model reactivity by the choice of λ. However EWMArequires the need to set or fit the parameter λ. A GARCH approach mayalso be used, which may provide fast reactivity in periods of crisis,however no clear gain compared to EWMA to justify the model complexity.A dual lambda approach may offer fast reactivity when necessary (e.g.,in periods of crisis) and would prevent margin from shrinkingconsecutively to a set of small returns (e.g., in calm period). DualLambda approach requires input or calibration of two decay parameters λand may be too conservative sometimes. Similarly, volatility flooringapproach may provide anti-procyclicality but may result in veryconservative (e.g., high) margins as well.

TABLE 3 Overview of methodologies for use in variance scaling MethodPros Cons Dual Lambda Fast reactivity in periods of Two parameters toset crisis Conservative measure: Prevents margin from Likely failing ofthe Kupiec shrinking consecutive to a test set of small returnsVolatility Anti-procyclicality Can potentially lead to very Flooringconservative (high) margins EWMA Fast Computation Need to set of fit theControl of the model parameter λ reactivity by the choice of λ GARCHFast reactivity in periods of More complex than crisis EWMA with similarresults Account for volatility clustering

The dimension reduction module 232 may be used to reduce thedimensionality (e.g., number of risk factors) when computing a marginrequirement for a portfolio of financial products and/or to present suchto a user via a user interface screen. The variance scaling module 234may be used to process a dynamic volatility model to scale a number ofnew risk factors obtained from dimension reduction module 232 and/or topresent such to a user via a user interface screen. The dynamicvolatility model processed by the variance scaling module 234 may beused to provide a sound set of innovations for simulations and toprovide a forward looking volatility measure suited for marginmeasurements. When considering a time series r(t′) of returns, whetherusing scores or using another time series, the innovations may bedefined as

$\begin{matrix}{{{{innovations}\left( t^{\prime} \right)} = \frac{r\left( t^{\prime} \right)}{\sigma \left( {t^{\prime} - 1} \right)}},} & (58)\end{matrix}$

where σ(t′−1) denotes the volatility estimate of the previous day. Theone day time lag is introduced to account for the forecast uncertaintyof volatility. A good volatility model may remove the dynamics of theindividual scores, such as the volatility clustering as shown in FIG.31. Further, successful volatility models fulfill the operationalrequirement of the margin measure (e.g. smoothness of the margins).Further, forward looking volatility will be used in the filteredhistorical simulation to reinsert the desired volatility to the sampleof uncorrelated innovations. The volatility models are always trained onthe expanding windows, starting at the beginning of each observationwindow being considered. Practically, a “burning” of 50 days may be usedto avoid any instability caused by the first volatility estimate, as canbe seen in FIG. 49.

The variance scaling module may process a score, an error term or a timeseries of parameter changes, where an exponentially weighted average(EWMA) volatility module may be defined as

$\begin{matrix}{{\sigma_{r}(t)} = {\sqrt{{\lambda_{EWMA}{\sigma_{r}^{2}\left( {t - 1} \right)}} + {\left( {1 - \lambda_{EWMA}} \right)\left( {{r(t)} - {(r)_{\lambda}(t)}} \right)^{2}}}.}} & (59)\end{matrix}$

λ_(EWMA) is the decay parameter, r(t) is the time series (e.g. scores,error term, parameter changes, factor loadings), and (r)_(λ) is theexponentially weighted estimate of the mean. A de-meaning procedure isonly used in the estimation of the time-varying volatility by complyingto the second term in the variance formula var(X)=E(X²)−E(X)². However,note that no model for the mean, such as an AR process, is used. Thevolatility estimate is utilized only for filtering purpose, as the meanmodel may not be required because that would require an extra model andparameterization to measure the mean and de-meaning would require toenter the mean value back in both the margin and the backtesting engine.Therefore the process for the average is

(r)_(λ)(t)=(1−λ)τ(t)+λ(τ)_(λ)(t−1)   (60)

and the time window is always expanding.

The parameter λ_(EWMA) usually ranges from about 0.94 to about 0.97.Values larger would fail to capture crisis as the static case is reachedwith a lambda of 1. We adopt an offset window (e.g., a burn in window)of 50 days to avoid finite size effects in the EWMA process. A seedvalue for the EWM average is the first return value and zero for the EWvariance. Expanding the recursive definition (59) of the EWMA volatilitymodel one observes that the model assigns the weights

ω_(n)={(1−λ), λ(1−λ), λ²(1−λ), . . . , λ^(N)(1−λ)} (61)

to the N historical dates, backwards from the most recent date. Thefinite sum of weights is

$\begin{matrix}{{{\sum\limits_{n = 1}^{\infty}\omega_{n}} = {\left( {1 - \lambda} \right) \cdot \left( {1 + \sum\limits_{1}^{N - 1}} \right)}},} & (62)\end{matrix}$

where,

$\begin{matrix}{\sum\limits_{1}^{N - 1}{= {\sum\limits_{k = 1}^{N - 1}{\lambda^{k}.}}}} & (63)\end{matrix}$

in the limit of N→∞ the sum of weights (62) is equal to unity. Forfinite and not large N, the sum of weighs may deviate from unity andleads to an underestimation of the volatilities. To correct this finitesize effect, (I−λ) may be replaced in (59) by

$\begin{matrix}\left. {1 - \lambda}\rightarrow{\frac{1}{1 + \sum\limits_{1}^{N - 1}}.} \right. & (64)\end{matrix}$

Consequently, for the coefficient λ equation 59,

$\begin{matrix}\left. \lambda\rightarrow{\frac{\sum\limits_{1}^{N - 1}}{1 + \sum\limits_{1}^{N - 1}}{\overset{\text{?}}{= {\lambda \; \frac{1 + \sum\limits_{1}^{N - 2}}{1 + \sum\limits_{1}^{N - 1}}}}.}} \right. & (65) \\{\text{?}\text{indicates text missing or illegible when filed}} & \;\end{matrix}$

In the implementation the initial window (e.g., burn in window) may beset to a number in the range of about 50 to about 100 business days forthe volatility estimate to avoid statistical effects due to the shortlength of the time series. Using a burn-in window the replacements inEq. (64) and (65) have no practical effect for the EWMA volatilityestimate. FIG. 49 shows a EWMA 0.94 on the first two PCs of crude oil.This figure is to be compared with the exact same plot, but using aGARCH model (FIG. 49). The volatility clustering of the scores, seen inFIG. 34, is almost disappeared as indicated by the autocorrelation plotof the squared innovations of the first component, as shown in FIG. 49A.

In some cases, a GARCH(1,1) model may be considered as candidate for thedynamic volatility model for use by the variance scaling module 234. TheGARCH tends to provide slightly larger volatility spikes. At the peak ofthe energy crisis in 2008, the EWMA first score volatility reaches 0.33,while the GARCH model reaches 0.36. By construction, the GARCH modelcaptures better the volatility clustering contained in the scores timeseries, yet the difference for the first lag is negligible. The biggestdifference is observed for the day 2 lag. As such, the advantagesprovided by using the GARCH process may provide more complexity than isnecessary with regards to its value added in terms of risk figures.

In some cases, the variance scaling module 234 may use a dual lambdaapproach that may include using two lambda parameters, λ_(fast) andλ_(slow), instead of one. Both lambda parameters are used in the EWM Varsetup (59) for variance estimation. The innovations are derived usingλ_(fast) solely, optimized for a daily horizon. However, the maximum ofthe two volatility estimates is taken when the innovations aremultiplied by the volatility forecast to find the P&L simulations. FIG.50A represents this interplay of two the parameters for the first scoreof crude oil. The dual lambda approach has several consequences, such assmoothing of the margins, high reactivity to crisis and slow reactivityto periods of calm. For example, the post-crisis margins are usuallyscaled using the slow lambda instead of the fast ones, making the decaymore progressive and smoother. Further, the dual lambda approach allowstaking a highly reactive fast lambda. Also, in periods of relativestability, such as observed in 2006 and 2013, the margins get driven bythe slow lambda, providing a slow decay towards lower margins. In thissense, the dual lambda is a first step towards anti-procyclicalitycondition, yet it does not prevent the margins from shrinking tovanishing.

The volatility flooring is an extra layer added on top of the volatilitymodel used, meant to add robustness and stability to the margins, and toprevent pro-cyclicality. At the generic simulation step, the volatilityused to simulate a given score or error term is floored by the nthvolatility quantile in a growing window as measured within the dynamicmodel.—In some cases, a configuration for which ICE Brent Crude firstscore volatility may be floored for a simulation. In an example, thefloor may be set to the 10% quantile, preventing the margins to becometoo low in late 2013.

FIG. 51 illustrates the historical value of the floor through time forthe first two scores of crude oil. As the quantile is measured on agrowing window, the learning effect results in a high stability of thefloor, making it nearly static. As the scores, and in particular thefirst one, are driving about 99% of the generics variance, to floor themhas a very strong impact on the risk measures and make the model veryconservative. For this reason, it is recommended to use a very lowquantile (e.g. 10%). FIG. 52 illustrates the sensitivity of the quantileparameterization when applied to the first score of ICE Brent. Asillustrated by the fast evolution of the floor at the beginning of thesimulation, the procedure requires a significant time history to beefficient.

The error term does not drive the variance of the generics, however itplays a significant role for the risk of intra-curve spreads such ascondors and butterflies. For intra-curve spreads very high correlationsbetween consecutive generics can make the margin measures very small.These correlations are described by the scores and residual terms. Note,that the PC analysis leads to uncorrelated scores but the genericcontracts can nevertheless be highly correlated. Therefore the errorterms do have significant contributions to the correlations of genericsand flooring the error terms influences the backtesting results for theintra-curve spreads. From these practical considerations and inspectionof backtesting results a stronger floor is recommended on the errorterm, i.e. around 40%.

The covariance scaling module 236 may be used to implement a multi-curvecorrelation model. Multiple models have been investigated to describecorrelation structures, where EWMC and DCC methods may be applied toboth intra-curve correlations and to describe correlations among scoresat inter-curve levels. The EWMC method may provide fast computationand/or control of the model reactivity by the choice of parameterlambda. However, this necessitates that a lambda value is set or fitbefore computing a margin value. A DCC method may provide a data drivenestimated correlation dynamic and may be more flexible, however themodel complexity is increased. Global PCA models may be used which mayprovide strong dimensional reductions and less inter-curve correlationsthat need to be captured through explicit covariance models, however,the results may be difficult to interpret and global PCs may not bestable.

TABLE 4 Overview of methodologies for use in covariance scaling MethodPros Cons EWMC Fast computation Need to set of fit λ Control of themodel reactivity by the choice of parameter λ DCC Data driven estimatedcorrelation Model complexity dynamic More flexible correlation dynamicGlobal PCA Strong dimensional reduction Difficult to interpret Lessinter-curve correlation has Global PCs are not stable to be capturedthrough explicit covariance models

The methodology for dimension reduction of the curves and variancescaling has been discussed above. After the described procedures thereturns of the single curves are projected to volatility scaled scores(innovations) and volatility scaled errors. Those innovations howevermay still contain correlations, especially on the inter-curve level. Themost appropriate model to capture these effects will be identified amongthe following approaches. The first approach is to describe each curveseparately and inter-curve correlation effects are implicitly andstatically captured through resampling of time-aligned innovations andresiduals. The second approach is to describe each curve separately anddynamic inter-curve correlation effects are explicitly captured througha model of the correlation between normalized principal components(multivariate EW correlation/DCC). Static correlation effects areimplicitly captured through resampling of time-aligned innovations andresiduals. The third approach is to jointly describe the curves byperforming a global PC rotation and then modeling the new risk factorsusing the same techniques used for single curves. The first approachassumes no dynamics in the correlation structure. Results on structuralbreaks, exponentially weighted correlation and constant conditionalcorrelation (CCC) tests do not support this assumption.

As the EWMC constitutes a weighted sum (with positive weights) of thetime series, instabilities of the correlation matrix (for examplenon-positivity) are excluded. Therefore the first model will beconsidered as benchmark model. The second approach captures dynamic incorrelations but can suffer from dynamic stability issues. A typicalsign is the occurrence of not well defined (not positive definite)correlation matrices. The third approach is an extension of theconsidered single curve methodology, however it suffers from thedifficult interpretability and instability of the considered factors.

The EWMC model corresponds to the iterative relation

C(t)=λC(t−1)+(1−λ) R (t) R ^(T)(t)   (69)

where C(t) is the covariance matrix, R(t) is a vector of time series andR^(T)(t) is given by

R (t)=R(t)−(R)(t)   (71)

where (R)(t) is the EWMA determined by

(R)(t)=λ(R)(t−1)+(1−λ)R(t)   (71)

Equation (69) assumes the same dynamic for both variances andcorrelations. For the practical implementation the identities (64) to(65) apply also for the EWMC computation. This considered model isslightly more flexible, indeed the diagonal part of the covariancematrix is estimated using an EW variance estimation with different λ asdescribed. Whereas the estimation of the covariance matrix of equation(69) focuses exclusively on the off-diagonal elements. Therefore, justthe correlation matrix Corr associated with the covariance matrix C iseffectively used to filter the scores innovations I_(t)=[I₁; . . . ,I_(N)] of the N modeled curves.

UI _(t)=chol(Corr)_(t−1) ⁻¹ ·I _(t)   (72)

where UI_(t) are the uncorrelated innovations. These are historicallyresampled during the FHS procedure. For risk management purposes, thereis a tradeoff between control and stability of the model and the need ofi.i.d residuals for the subsequent filtered historical simulations. Insome cases, two models may be compared to capture dynamic correlation.The first model aims at stability and robustness: a EWMC model with afixed lambda parameter, chosen on historical data. The second modelfocuses on flexibility and optimal filtering a DCC model retrained ateach time step based on maximum likelihood estimation (MLE). There isalso the somehow intermediate approach of recalibrating a EWMC model ateach time step. This is not considered since for this model theestimation of lambda does not decouple correlation and covarianceeffects. Therefore it is inconsistent with the use of the correlationmatrix scaling as in Eq. (72), but may be useful in other cases.

The covariance scaling module 236 may process instructions to implementa dynamical conditional correlation (DCC) model, as opposed to the EWMCapproach, where the DCC model may operate under an assumption ofdifferent dynamics for each variance and a different, but unique,dynamic for the correlation structure. For example,

C(t)=D(t)Q*(t)D(t)   (73)

Q*=diag(Q(t))^(−1/2) Q(t)diag(Q(t))^(−1/2)   (74)

where C is the covariance matrix and D(t) is given by

D(t)=diag(d ₁(t), . . . , d _(N)(t))   (75)

The diagonal entries in (75) are defined by:

d _(i)(t)=√{square root over (c_(ij)(t))}, i=1, . . . , N   (76)

where, c_(ij)(t) are the elements of the covariance matrix C(t). TheModel dynamics may be given in terms of the matrix Q(t) as

Q(t)=αQ(t−1)+βR* ^(T)(t−1)R*(t−1 )+(1−α−β) Q   (77)

where α and β are parameters and Q is the normalized unconditionalvariance matrix of R_(i)(t). The time series R_(i)*(t) is the(normalized) unconditional variance matrix of R_(i)(t).

R* _(i)(t)=R _(i)(t)/σ_(i)(t)   (78)

where σ_(i)(t) are univariate estimated volatilities of R_(i)(t). Thefirst step is the estimation of the univariate volatilities σ_(i) bycalibrating GARCH(1,1) processes to the time series Ri(t). Second, thematrix Q is the static estimate of the covariance matrix normalized bythe static volatilities σ_(i,) e.g.,

Q _(ij)=Cov(R _(i) , R _(j))/(σ _(i) σ _(j))   (79)

where σ _(i) is given

$\begin{matrix}{{\overset{\_}{\sigma}}_{i}^{2} = {\frac{1}{N - 1}{\sum\limits_{n = 1}^{N}\; \left( {{R_{i}\left( t_{n} \right)} - {\overset{\_}{R}}_{i}} \right)^{2}}}} & (80)\end{matrix}$

where R _(I) is the mean. Third, the parameters α and β are found by amaximum likelihood estimation (MLE) fitting procedure, where thelikelihood function reads:

$\begin{matrix}{{L\left( {\alpha,\beta} \right)} = {\frac{1}{2}{\sum\limits_{t}\; {\left( {{\ln \left( {2\; \pi} \right)} + {\ln \left( {\det \mspace{11mu} {Q(t)}} \right)} + {\sum\; {{R^{*T}(t)}{R^{*}(t)}\text{/}{{diag}\left( {Q(t)} \right)}}}} \right).}}}} & (81)\end{matrix}$

Fourth, the Cholesky decomposition of the DCC correlation matrixQ_(i)*(t) is computed and inverted. Finally, the resulting matrix isused to generate uncorrelated innovations

UI _(t)=chol(Q*(t))_(t) ⁻¹ ·I _(t)   (82)

Note that the time convention in the definition of UI_(t) for the DCCmodel is different from the corresponding one for the EWMC model inequation 72. The MLE fitting procedure of this model can be decoupled ina variance estimation procedure and a pseudo-correlation estimation onvolatility filtered returns. This makes the calibration of this modelconsistent with the subsequent use of correlation scaling suggested inEquation (72). Furthermore, this model has the advantage of using aGARCH-like dynamic for the covariance, which is more flexible than thedynamic assumed by using exponential weighted estimation. The modelprovides therefore a suited calibration procedure for optimalcorrelation filtering as well as sufficient flexibility to capturechanges in the correlation dynamics. The downside is that the dynamicsof the parameters and can be substantial and lead to model instability.

As a test case, the DCC and EWMC models are trained on the pair ICEBrent crude/WTI Crude oil. The two parameters in the used time serieswere both always positive, and their sum was slightly lower 1,indicating a satisfactory fit. The stability of the fit (based on agrowing time window) is supporting a DCC approach. FIGS. 53A and 53Bshow an illustrative comparison of the resulting correlation of thefirst generics of the two commodities discussed above. For example, FIG.53A illustrates front month correlations of the curve spreads from anEWMC 0.97 model. FIG. 53B illustrates front month correlations of thecurve spreads from a DCC model. The break in correlation of January 2011results in both cases in a strong drop. With EWMC, it decreases to52.5%, and then fluctuates around 80%. While using DCC, it falls to 65%,and then fluctuates around 85%. In the context of a spread product, theEWMC tend to give conservative estimates of the correlations. Thisobservation is confirmed by the large scale results.

The VaR estimation module 238 may process instructions to facilitate aforecast simulation of returns is necessary for obtaining an estimate ofthe value at risk (VaR). In order to generate scenarios we use filteredhistorical simulations (FHS). This approach combines two ideas, namelysampling of historical residuals and scaling of residuals by a modeldriven parameter such as the forecast volatility or correlation.

Given the raw futures price data F_(Ti)(t), where Ti is the expirationdate of the ith contract, one finds generic contract return R_(n) ^((g))(t), where n is a generic number according to its time-to-maturity. Incontrast to interpolation, rolling does not introduce artificial pricedata. However, with a rolled contract the time-to-maturity is notstrictly fixed as is the case using interpolated data points.

Second, the return surface, i.e. the returns depending ontime-to-maturity and observation time, is dimensionally reduced. Thereare two possible reductions: principle component analysis (PCA) orparametric models. The following shows a PCA approach. Having derived anestimate of the covariance matrix C (by EWMC or multivariate GARCHmodel) we are diagonalizing C as

C(t)=PC(t)Λ(t)PC ^(T)(t)   (83)

to obtain the principle components. Using the orthogonal matrixPC=[PC₁(t), . . . , PC_(N)(t)], the principle components may be foundusing

$\begin{matrix}{{{{pc}_{k}\left( {t,t^{\prime}} \right)} = {\sum\limits_{j = 1}^{N}\; {{{PC}_{j\; k}(t)}{R_{j}\left( t^{\prime} \right)}}}},} & (84)\end{matrix}$

Where we distinguish the training window (t′) and the derivation date ofprinciple components (t). Multiplying the last equation with the PCmatrix we can relate the scores to the original return time series.

$\begin{matrix}{{{R_{i}\left( t^{\prime} \right)} = {{\sum\limits_{k = 1}^{n}{P{{C_{ik}(t)} \cdot {{pc}_{k}\left( {t,t^{\prime}} \right)}}}} + {\epsilon_{i}\left( t^{\prime} \right)}}},} & (85) \\{{\epsilon_{i}\left( t^{\prime} \right)} = {\sum\limits_{k = {n + 1}}^{N}{P{{C_{ik}(t)} \cdot {{{pc}_{k}\left( {t,t^{\prime}} \right)}.}}}}} & (86)\end{matrix}$

where we have noted the two time arguments explicitly. Retaining only asmall number of scores, n is usually fixed between n=3 and n=5.Additionally, N-n residual scores are kept in error term ∈(t′). Nomatter if non-updating or updating of scores is selected, one finds withthe few time series for the scores and the remaining residuals. Thetotal number of time series equals the number of generic futures.

Third, innovations I may be defined as:

$\begin{matrix}{{I_{k}\left( {t,t^{\prime}} \right)} = \frac{p{c_{k}\left( {t,t^{\prime}} \right)}}{\sigma_{I}^{k}\left( {t^{\prime} - 1} \right)}} & (87)\end{matrix}$

where σ_(I) ^(k)(t′−1) is the volatility of the kth score pc_(k)(t, t′)on the previous business day. Different volatility models can be used toestimate σ_(I). Again, a choice may be made among EWMC or multivariateGARCH, dual-lambda and the option of volatility flooring. However, notethat volatilities limited by flooring are not used to standardizeinnovations but only to scale back the standardized innovations.

Fourth, filtering the correlation structure on the inter- and intracurvelevel. The correlation matrix may be estimated using one of thefollowing models: EWMC, DCC, and global PCA. Each of the models leads toa correlation matrix Corr that is used to generate uncorrelatedinnovations by

UI _(t)(t,t′)=chol(Corr)_(t′−1) ⁻¹ ·I(t,t′)   (88)

where chol denotes the Cholesky decomposition and UL_(t)(t, t′) are thetime series of uncorrelated innovations at derivation date t. Note, thatsimilar to the volatility scaling step, this decorrelation step is doneon each day of the look-back window using that day's correlation matrix.

After the correlation scaling we are left with a set of uncorrelatedinnovations UI and variance scaled residual terms (85). The latter read

$\begin{matrix}{{z_{i}\left( {t,t^{\prime}} \right)} = \frac{p{c_{i}\left( {t,t^{\prime}} \right)}}{\sigma_{ɛ}^{i}\left( {t^{\prime} - 1} \right)}} & (89)\end{matrix}$

where σ_(ε) ^(i)(t′) is the variance estimate of pc_(i)(t, t′). Theindex i in Equation (89) selects only PC scores which contribute to theerror term in Equation (85). The time-aligned uncorrelated innovationsUI and residuals z_(i) are assumed to i.i.d and are sampled for scenariogeneration. In this way the empirical return distributions is accountedfor and we obtain the simulations UI_(sim)(t) and variance scaledresidual simulation z_(sim)(t). The simulations of uncorrelatedinnovations are scaled back by

pc _(sim)(t)=diag(σ(t))·chol(Corr(t))·UI _(sim)(t)   (90)

The simulated scaled residual terms z_(sim) ^(i)(t)are multiplied by thecorresponding standard deviation σ_(ε) ^(i)(t) and sampled in line withthe innovations. Explicitly

ε_(sim) ^(i)(t)=z _(sim) ^(i)(t)·σ_(ε) ^(i)(t),i=n+1, . . . N   (91)

Furthermore the simulations of scores and resealed residuals aretransformed back to returns as in Eq. (85). Summing both terms leads tothe simulated returns:

r _(sim) ^(i)(t)=Σ_(k=1) ^(n) PC _(ik)(t)·pc _(sim) ^(k)(t)+Σ_(k=n+1)^(N) PC _(ik)(t)·ε_(sim) ^(k)(t)   (92)

Using the return simulations r_(sim) ^(i)(t) we obtain P&L simulation onthe portfolio level by

$\begin{matrix}{\mspace{79mu} {{{{P\&}{L_{sim}(t)}} = {\sum\limits_{i = 1}^{n_{L}}\; {\omega_{i}\left( {{e\text{?}F^{i}\text{?}(t)} - {F^{i}\text{?}(t)}} \right)}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (93)\end{matrix}$

where n_(L), ω, and F^(i) _(τ)(t) are the number of legs, the ith leg'sweight (position size) and the ith futures price at time t and ttm τ,respectively. Finally, the margin VaR is estimated as the a quantile ofthe simulated sample P&L_(sim).

Dual Lambda and volatility flooring may be performed by the VaRestimation module 238. The default exponentially weighted estimate withparameter λ for the variance of the time series r(t) reads

σ_(EWM) ²(t)=λσ_(EWM) ²(t−1)+(1−λ)r ²(t)   (94)

In the dual-lambda approach one considers two parameters λ_(fast);λ_(slow) and the corresponding processes of the form (94). Hereattributes “fast” and “slow” should indicate how reactive the varianceestimate is. These values may be chosen from a range of values fromabout 0.9 to about 1, where typical parameter values are

λ_(slow)=0.99   (95)

λ_(fast)=0.94   (96)

In general, the final volatility estimate for the dual-lambda approachmay be found by

σ(t)=max(σ_(slow)(t), σ_(fast)(t))   (97)

However, in deriving the innovations by Eq. (87) σ_(fast)(t) is used.The result of Equation (97) multiplies the simulations of theinnovations and thereby influences the margin estimation.

Volatility flooring introduces a lower limit to the volatilityestimation. The estimate for σ_(Model)(t), as computed by processes suchas (94) may be compared to fixed nth, quantile Q_(n) of the historicalvolatility. Specifically, the sample of historical volatilities {σ(s)with s≤t} within a growing estimation window is used to estimate thequantile

σ_(q) _(n) (t)={circumflex over (Q)} _(n){σ(s)|s≤t}   (98)

The floored volatility is found as

σ(t)=max(σ_(Model)(t),σ_(Q) _(n) (t))   (99)

As in the dual-lambda, in the practical implementation the resultingvolatility of Equation (99) is only used on the simulation level torescale the simulation of innovations as in Eq. (90). The flooring isnot employed in deriving the innovations. Statistically smallvolatilities may be avoided by following the flooring strategy and mayintroduce additional stability to margins.

The VaR estimation module 238 may choose the sample to use for VaRestimates to be equal to the entire currently available N_(Hist)historical innovations. This option has the advantage of providingstable VaR time series. However, a more conservative approach whichaccounts for Monte-Carlo VaR estimation errors can be considered. Inthis case the VaR estimation is based on a bootstrapping procedure. Thisconsists first in generating N_(Btstr) samples (e.g., set N_(Btstr)=100)of N_(Hist) innovations obtained by resampling with replacementhistorical innovations and calculating for each sample a VaR estimatesVaR_(n) (n=1 . . . N_(Btstr)). The final VaR is obtained as the5%-quantile of the obtained empirical VaR distribution.

In some cases, the margin calculator 240 may process a backtestingprocedure where various combinations of instruments may be simulatedusing the proposed parameterization. The selection criteria may be oneor both qualitative and quantitative.

In some cases, the margin calculator 240 may perform a quantitativetest, such as an unconditional coverage test. The goal of this test maybe to measure whether it is statistically reasonable that the reportedvalue-at-risk is violated more (or less) than α×100% of the time. Tothis end, the Kupiec test may be used. A violation may be described as aBernoulli trial:

$V:=\left\{ {\begin{matrix}{1,} & {{if}\mspace{14mu} {VaR}_{\alpha}\mspace{14mu} {exceeded}} \\{0,} & {else}\end{matrix}.} \right.$

These heuristics give rise to the following null hypothesis: Vt followsthe binomial distribution with parameters N and q=1−α. Thus, the testconsists of evaluating if the estimated parameter

$\overset{\hat{}}{q} = \frac{n_{V}}{N}$

is significantly different from the value 1−α with a certain confidencelevel (CL), n_(V) being the total number of var violations observed onthe sample and N denotes the sample size. We reject the null hypothesisif {circumflex over (q)} is significantly different from 1−α. Theconfidence interval [Q_(lb);Q_(ub)] at confidence level (e.g., CL=99%)for the estimated parameter {circumflex over (q)} is found using theClopper Pearson method [6]. For the upper bound of the confidenceinterval one has

$\begin{matrix}{{v_{1}^{ub} = {2\left( {n_{V} + 1} \right)}},} & (103) \\{{v_{2}^{ub} = {2\left( {N - n_{V}} \right)}},} & (104) \\{{X_{ub} = {F^{- 1}\left( {\frac{CL}{2},v_{1}^{ub},v_{2}^{ub}} \right)}},} & (105) \\{Q_{ub} = {\frac{v_{1}^{ub}X_{ub}}{v_{2}^{ub} + {v_{1}^{ub}X_{ub}}}.}} & (106)\end{matrix}$

The lower bound reads

$\begin{matrix}{{v_{1}^{lb} = {2\; n_{V}}},} & (107) \\{{v_{2}^{lb} = {2\left( {N - n_{V} + 1} \right)}},} & (108) \\{{X_{lb} = {F^{- 1}\left( {{1 - \frac{CL}{2}},v_{1}^{lb},v_{2}^{lb}} \right)}},} & (109) \\{Q_{lb} = {\frac{v_{1}^{lb}X_{lb}}{v_{2}^{lb} + {v_{1}^{lb}X_{lb}}}.}} & (110)\end{matrix}$

The transformation in (106) and (110) may be used to utilize the inverseF-distribution. Several formulations of the Clopper Pearson method maybe used. Alternatively, a β-distribution may be used directly to findthe lower and upper bounds. The likelihood ratio for the Kupiec testreads:

LR _(kupiec)=−2ln(α^(NV)(1−α)^(N−n) ^(V) )+2ln(q ^(n) ^(V) (1−q)^(N−n)^(V) )   (111)

where α is the level in VaR_(α). The likelihood ratio is asymtoticallyx_(i) ² distributed with one degree of freedom. Therefore the p-valuefor the Kupiec test may be given by:

p _(kupiec)=1−X ₁ ²(LR _(kupiec))   (112)

In some cases, margin calculator 240 may perform an independence test,where the goal of the independence test may be to assess thesignificance of violation clustering throughout the testing set. A goodmargin model may avoid clustering of the violations, even if theindependence test is successful.

The Christoffersen test may be used. The core idea is to investigatewhether there is a time dependency (as measured by the time elapsedbetween two consecutive violations) in the testing sample. Under theviolation independence assumption, no time dependency should bedetected. Specifically, let v=(v₁, . . . , v_(n)) ∈N_(n) be the vectorof time indices between two VaR_(α) violations, where v_(1 is) the timeof the first violation. The likelihood ratio for each violation is givenby:

$\begin{matrix}{\mspace{79mu} {{{LR}_{i} = {{- 2}\; {\log\left( \frac{{p\left( {1 - p} \right)}\text{?}}{\left( \frac{1}{v\text{?}} \right)\left( {1 - \frac{1}{v\text{?}}} \right)\text{?}} \right)}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (113)\end{matrix}$

where I=(1, . . . n). Under the null hypothesis of independent VaR_(α)violations:

$\begin{matrix}{\mspace{79mu} {{LR} = {\sum\limits_{i = 1}^{n}\; {{\left\lbrack {{- 2}\; {\log\left( \frac{{p\left( {1 - p} \right)}\text{?}}{\left( \frac{1}{v\text{?}} \right)\left( {1 - \frac{1}{v\text{?}}} \right)\text{?}} \right)}} \right\rbrack.\text{?}}\text{indicates text missing or illegible when filed}}}}} & (114)\end{matrix}$

is x_(i) ² distributed with n degrees of freedom. The p-value for theChristoffersen test therefore reads:

p _(Chris)=1−X _(n) ²(LR)   (115)

The outcome of these two tests is measured by the resulting p-value. Thecolor convention may be specified as a function of the confidence levelconsidered, and is described in FIG. 54.

The margin module may provide an additional quantitative indicators,such as a mean VaR violation rate that may be defined:

$\begin{matrix}{{{{mean}\; {VaRRate}} = \frac{n_{V}}{N}},} & (116)\end{matrix}$

with n_(v) the total number of violations observed on the simulationsample of size N (the training set may be excluded) and may be expressedin percentage. This indicator may be key to understand the impact of afailed Kupiec test, such as whether the risk is over-estimated orunder-estimated

A mean break size indicator may be defined as a mean ratio of theviolations by the corresponding VaR number:

$\begin{matrix}{\mspace{79mu} {{{meanBreakSize} = {\frac{1}{n_{V}}{\sum\limits_{v \in V}^{n_{V}}\; \frac{{PL}\left( t_{v} \right)}{{VaR}_{\alpha}\left( {t\text{?}} \right)}}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (117)\end{matrix}$

with the t_(v) sample time of each of the n_(v) violations contained inthe violation set V and may be expressed in percentage.

A maximum margin increase indicator may give an overview of thesmoothness of the margin curve and may be expressed in percentage andmay be defined by:

$\begin{matrix}{\mspace{79mu} {{maxMarginIncrease} = {\max \text{?}{\left( \frac{{VaR}_{\alpha}\left( {t + 1} \right)}{{VaR}_{\alpha}(t)} \right).\text{?}}\text{indicates text missing or illegible when filed}}}} & (118)\end{matrix}$

In some cases, the margin calculator 240 may determine a maximum numberof breaks within a one year moving window (e.g., a 252 moving window)that may be used on the training set. The maximum observed is reported.This indicator may be useful to understand a failing independence test.Moreover, a plot may be provided in the backtesting report that mayillustrate when exactly the maximum is reached throughout the backtestperiod. An example is shown in FIG. 55.

Because the numerical procedures may need to be supported in validatinga module, therefore one or more qualitative inspection methods may beprovided to provide a visual check of the model. One or more inputs maybe provided to the user to allow an indication of whether or not thetest has been passed to be used by the margin calculator 240 incomputing a margin or indicating that the test has been failed. Forexample, different scenarios may cause the margin calculator 240 toprovide one or more visual indicators to a user via a user interfacescreen in cases including a failed Kupiec test due to riskover-estimation (e.g., is the over-estimation acceptable) or a failedChristoffersen test due to a local clustering effect. ThisChristofferson independence test is strongly sensitive not only to theclustering of violations, but also to the number of violations. Theresults should therefore be subject to interpretation via a userinterface screen. Other user interface screens may be used to presentresults for illiquid curves, smoothness of the margins and/or a level ofanti-procyclicality. Some tools (e.g., volatility flooring, dual-lambda)may allow the model to be made more robust regarding economic cycles.However, they tend to make Kupiec tests fail. In some cases, results atthe back end of the curve may be presented to a user via a userinterface screens. The back-end of the curve can be subject to lowliquidity and artificial data creation (e.g., a condor on crude oil inthe back end of the curve, which may result in a trinomial distribution,plus or minus 1 cent, or 0). As the model may not be applicable ondiscrete distributions, the results of the Kupiec and Christoffersentests should be subject to caution when applied in these cases.

FIG. 3 shows an illustrative method performed using computer-executableinstructions when processed by a clearinghouse computing systemaccording to aspects of this disclosure. For example at 310 theclearinghouse computing system 210 may retrieve informationcorresponding to a portfolio of financial products including a pluralityof futures products. To determine a margin contribution for each of theplurality of futures products, the time series generator 230 may be usedto construct generic time series for each futures product and/or familyof futures products as discussed above. The rolling procedure may bepreferred over the interpolation and has an advantage of allowingone-to-one mapping with the existing future. Empirical statisticalanalysis has shown that the time to maturity effect for the front monthis small. In the back-end of the curve, an extrapolation technique maybe used to fill the missing data and thus generate generics for thesefar months. Linear extrapolation is used on the price level. If a timeseries is missing, an extension model is designed to generate artificialdata. The time series generator uses the maximum data available and doesnot truncate long-time-series because some of the curves have limitedtime-range. Specifically, a Student's t-distribution inverse conditionalexpectation formula may be used.

At 320, the dimension reduction module 232 may process instructionsstored in the memory 214 to perform dimension reduction for the generictime series generated by the time series generator. Here, every listedproduct price and maturity is considered a “dimension.” In doing so, alarge data set is created when every dimension of a large suite ofproducts must be analyzed to determine margin. By utilizing thePrincipal Component Analysis (PCA) technique, the term structure curvedynamics can be captured by a few factors instead of the full set ofmaturities. The reduced set of factors identified by PCA may then beused to describe the correlation dynamic. The remaining factors will bestill considered when doing the variance scaling for single product. Assuch, the large number of dimensions is reduced when facingmulti-products, while the volatility for single product is stillpreserved.

At 330, the variance scaling module 234 may perform variance scalingusing a dynamic EWMV (Exponentially Weighted Moving Variance) volatilitymodel. This EWMV model may be applied on both the major factorsidentified by PCA and the residuals to remove the volatility clusteringon the time series of data and provide an independently, identicallydistributed (i.i.d.) innovations for the historical simulations. Thesecond goal is to provide a forward looking volatility measure suitedfor margin measurements.

At 340, the covariance scaling module 236 may perform covariance scalingon a multi-product data set (e.g. a portfolio containing multipleproducts), the handling of correlation dynamics may be crucial forportfolio margining. A EWMC (Exponentially Weighted Moving Covariance)correlation model may be applied on the major factors identified by PCAfor all products, in order to scale the correlated innovations obtainedfrom step 2 to non-correlated i.i.ds, as well as to provide a forwardlooking correlation suited for margin measurements. In some cases, thecorrelation model may be applied only on the few selected PCs. Theassumption is thus that the intra- and inter-curve correlation dynamicsis captured by these few PCs for all curves. Tests have shown that theDCC model may be used, as opposed to an EWMC model, but the results showonly a marginal improvement with regards to the complexity of DCC andthe need to fit the parameters.

At 350, the VaR estimation module 238 may be used to generate theforecast simulation of returns to obtain an estimate of the VaR. Inorder to generate scenarios, the filtered historical simulation (FHS) isused. This approach samples the uncorrelated historical innovationsobtained and scales them by model driven parameters, namely, theforecast volatility and correlation. The scaled factors are thentransformed back to the price log returns which will be treated as theprice shocks and be applied at the portfolio level to obtain theportfolio shocks. The final margin value or VaR comes from a pre-definedpercentile of the portfolio shock distribution. In some cases, aFiltered Historical Simulation may be used based on the chosenvolatility model, where in case of a dual lambda model, only the fastlambda is used to derive the innovations. The obtained innovations(e.g., the expanding windows of all the data points available) for thescores are de-correlated using a Cholesky decomposition, and aftersimulation, they are resealed again using the correlation matrixobtained from the correlation model. Once the PCs and residuals aresimulated, they are finally rotated back to the return space using thePCs. Here, the residuals (e.g., the left scores) are also simulatedusing Filtered Historical Simulations, however they are not adjusted bya dynamic correlation model. The VaR estimate (e.g., margin value) canfinally be measured as the quantile of the obtained scenarios

At 360, the margin calculator may perform backtesting and/or othervalidation tests to ensure validity of the computed margin value. Insome cases, the margin calculator may sum a plurality of margin values(e.g., VaR values) computed for a subset of a portfolio, to generate amargin requirement for the entire portfolio.

Conclusion

The foregoing description of embodiments has been presented for purposesof illustration and description. The foregoing description is notintended to be exhaustive or to limit embodiments to the precise formexplicitly described or mentioned herein. Modifications and variationsare possible in light of the above teachings or may be acquired frompractice of various embodiments. For example, one of ordinary skill inthe art will appreciate that the steps illustrated in the illustrativefigures may be performed in other than the recited order, and that oneor more steps illustrated may be optional in one or more embodiments.The embodiments discussed herein were chosen and described in order toexplain the principles and the nature of various embodiments and theirpractical application to enable one skilled in the art to make and usethese and other embodiments with various modifications as are suited tothe particular use contemplated. Any and all permutations of featuresfrom above-described embodiments are the within the scope of theinvention.

1. A computing device comprising: a processor; and a non-transitorymemory coupled with the processor and storing instructions that, whenexecuted by the processor, cause the processor to: (a) generate, by atime series generator, a time series of pricing informationcorresponding to a financial product held in a portfolio and comprisinga plurality of factors contributing to dynamics of the time series; (b)calculate, by a dimension reduction module, a projection of the timeseries of pricing information which uses less than all of the pluralityof factors to represent the dynamics of the time series; (c) calculate,by a variance scaling module, a volatility normalization of theprojection to produce a plurality of curves; (d) determine, by acovariance scaling module, an inter-curve correlation and an intra-curvecorrelation between the plurality of curves; and (e) generate, by avalue at risk estimation module, an estimated value at risk based on theinter-curve and intra-curve correlations between the plurality ofcurves.
 2. The computing device of claim 1 wherein a calculated marginrequirement corresponds to the estimated value at risk generated by thevalue at risk estimation module.
 3. The computing device of claim 1,wherein the portfolio comprises two or more financial products, whereinthe non-transitory memory comprises instructions that, when executed bythe processor, cause the processor to: perform (a)-(e) for each of twoor more financial products in the portfolio; and calculate, by a margincalculator, a portfolio margin requirement based on the estimatedvalue-at-risk of for each of the two or more financial products.
 4. Thecomputing device of claim 1, wherein instructions cause the processor,as part of (a), to extrapolate, by the time series generator, one ormore missing data points.
 5. The computing device of claim 4, whereinthe one or more missing data points are at the tails of a pricing curveof the financial product.
 6. The computing device of claim 1, whereinthe projection comprises 80 percent fewer factors than the generatedtime series.
 7. The computing device of claim 1, wherein the projectioncomprises less than 10 factors.
 8. The computing device of claim 1,wherein (c) is performed using a dual lambda approach wherein avolatility of the scores is set to the maximum of a fast lambda EWMA anda slow lambda EWMA.
 9. The computing device of claim 1, furthercomprising a user interface device coupled with the processor andincluding a display device and a user input device, wherein thenon-transitory memory device stores instructions that, when executed bythe processor, cause the processor to: present, to the user via thedisplay device, at least one backtesting screen comprising a result froma test; and receive, via the user input device, an indication whether amodel for determining a portfolio margin requirement has passed aqualitative validation test.
 10. A computing system comprising: ahistorical pricing database storing pricing information for a pluralityof financial products; a computing device communicatively coupled to thehistorical pricing database and comprising: a processor; and anon-transitory memory coupled with the processor and storinginstructions that, when executed by the processor, cause the processorto: (a) generate, by a time series generator, a time series of pricinginformation corresponding to a financial product held in a portfolio andcomprising a plurality of factors contributing to dynamics of the timeseries; (b) calculate, by a dimension reduction module, a projection ofthe time series of pricing information which uses less than all of theplurality of factors to represent the dynamics of the time series; (c)calculate, by a variance scaling module, a volatility normalization ofthe projection to produce a plurality of curves; (d) determine, by acovariance scaling module, an inter-curve correlation and an intra-curvecorrelation between the plurality of curves; and (e) generate, by avalue at risk estimation module, an estimated value at risk based on theinter-curve and intra-curve correlations between the plurality ofcurves.
 11. The computing system of claim 10 wherein a calculated marginrequirement corresponds to the estimated value at risk generated by thevalue at risk estimation module.
 12. The computing system of claim 10,wherein the portfolio comprises two or more financial products, whereinthe non-transitory memory device comprises instructions that, whenexecuted by the processor, cause the processor to: perform (a)-(e) foreach of two or more financial products in the portfolio; and calculate,by a margin calculator, a portfolio margin requirement based on theestimated value-at-risk of for each of the two or more financialproducts.
 13. The computing system of claim 10, wherein instructionscause the processor, as part of (a), to extrapolate, by the time seriesgenerator, one or more missing data points.
 14. The computing system ofclaim 13, wherein one or more the missing data points are at the tailsof a pricing curve of the financial product.
 15. The computing system ofclaim 10, wherein the projection comprises 80 percent fewer factors thanthe generated time series.
 16. The computing system of claim 10, whereinthe projection comprises less than 10 dimensions
 17. The computingsystem of claim 10, wherein (c) is performed using a dual lambdaapproach wherein a volatility of the scores is set to the maximum of afast lambda EWMA and a slow lambda EWMA.
 18. The computing device ofsystem 10, further comprising a user interface device coupled with theprocessor and including a display device and a user input device,wherein the non-transitory memory device stores instructions that, whenexecuted by the processor, cause the processor to: present, to the uservia the display device, at least one backtesting screen comprising aresult from a test; and receive, via the user input device, anindication whether a model for determining a portfolio marginrequirement has passed a qualitative validation test.
 19. A methodcomprising: generating, by a time series generator, a continuous timeseries of pricing information corresponding to a financial product heldin a portfolio and comprising a plurality of factors contributing todynamics of the continuous time series; calculating, by a dimensionreduction module, a projection of the time series of pricing informationwhich uses less than all of the plurality of factors to represent thedynamics of the time series; scaling, by a variance scaling module, byperforming volatility normalization of the projection to produce aplurality of curves; determining, by a covariance scaling module, aninter-curve correlation and an intra-curve correlation between theplurality of curves; and generating, by a value at risk estimationmodule of the clearinghouse computer device, an estimated value at riskbased on the inter-curve and intra-curve correlation between theplurality of curves.
 20. The method of claim 19, comprising: performing(a)-(e) for each of two or more financial products in the portfolio; andcalculating, by a margin calculator, a portfolio margin requirementbased on the estimated value-at-risk of for each of the two or morefinancial products.
 21. The method of claim 19, comprising: validating,by the margin calculator, a margin generation model using one or more ofa qualitative backtesting method or a quantitative backtesting method.